Arthur Sherman - Chemical Vapor Deposition for Microelectronics (779637), страница 5
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Since p, T and density p are related by the equation of state, we canexpect p = p(x). However, the variation in density with x will be small, and itwill be reasonable to neglect it.As the reacting gas flows down the channel, it interacts with the channelwalls, decomposes, and leaves a film on these walls.
If the wall deposit is rapidand heavy, the reactants will deplete so that the gas composition will vary withx. Although this is a technologicarly important case, it requires a two-dimensional (partial differential equations) description. For the present problem, wewill assume that depletion is slow enough for us to neglect, and gas compositionwill not be a function of x.As in the classical Poisseulle flow, the y component of velocity will be zero,so that the overall mass continuity equation is identically satisfied.
For a steadystate flow, we can write the simplified governing equations describing thevelocity, temperature, and species conservation fields.Fundamentals of Thermal CVD19Momentum Conservation:(30).Q£dxwhere J.1 (gas viscosity) is a function of T and gas composition.Energy Conservation:nnL(31 )j=lwheredTPYJ.V y . c p . dyJJ+Lj=lWj cpoJTwJ.thermal conductivity (a function of T and gas composition)mass fraction of species jspecific heat at constant pressure of species jmolar production rate of species jmolecular weight of species jy component of diffusion velocity for the j species.Species Mass Conservation:(32)and there is one such equation for each species j.In order to complete specification of these equations, we have to expressthe diffusion velocity in terms of the species concentrations. We have,(33)where Vc y is a constant chosen to ensure the conditionnLj=lVy .
Y.J Joobtained by summing Equation (32) over all species, the mole fraction Xj isrelated to the mass fraction, Yjl byThe diffusion coefficient, Dj , refers to the diffusion of species j through the entire gas mixture. It can typically be evaluated approximately from the binary20Chemical Vapor Deposition for Microelectronicsdiffusion coefficients, which refer only to binary gas mixtures. lS The lattercan be calculated from rigorous kinetic theory.Similarly, the viscosity and thermal conductivity can be evaluated approximately with the help of kinetic theory arguments.
1S Finally, we need an equation of state relating p, p, and T. Assuming we are dealing with a mixture ofperfect gases, we havep(34)=p RTwhere R is the mixture gas constant which is equal to ~/w, with at the universalgas constant (1.987 cal/moleoK) and w the average molecular weight of the gas.In order to solve these equations, we have to be able to evaluate c:i)j, thespecies net production rate as a function of conditions and gas composition.If we assume only binary reactions and an Arrhenius temperature dependencefor the forward rate coefficients of such reactions, then we can express Wj ina reasonably simple form.First, let's choose the simple reaction of silane pyrolosis and solve oursimplified equations for this case.
Then, we haveso that there are three species to keep track of.Then, if we refer to them aswe havewhere k f and k r are the forward and reverse reaction rates of our one reactionequation, and [Xl], and [X 2 ] and [X 3 ] are molar concentrations. As shouldbe obvious, destruction of one SiH 4 molecule produces one Si H 2 and one H 2 .The forward rate coefficient is(36)kf= A exp [-E/RT]where A and E are experimentally determined constants. The reverse rate coefficient is related to the forward one at equilibrium bywhere K c is equilibrium constant in concentration units.
Since we are dealingFundamentals of Thermal CVD21with a quasiequilibrium, we will use this to determine k r . It is simpler to determine it from its pressure units form. The relationship between these forms is,for our case,PatmTK --(37)Pwhere Patm is atmospheric pressure, and Kp can be obtained from/;,5°6HO)Kp = exp ( --- ~RRT(38)where ~So is the change in entropy of the gases in our reaction in going fromreactant to products under standard state conditions (atmospheric pressure).Then LiH o is, similarly, the change in standard state enthalpy.
The standardentropies, enthalpies and specific heats at constant pressure are all tabulated inthe JANAF Table. 4 We can now express the species production rates asRT[XS"iH ] [X H ]2 0 2A exp[-E/RT]656HOJPatm exp [ -R- -- ~(39)andor replacing species concentrations by species mass fractions, this becomespA exp[-E/RTJ(40)In order to proceed with calculations, Cp for each species and ~Ho/RTplus ~So/RT can be expressed as functions of temperature using the JANAFTables. 4Finally, we have to define the proper boundary conditions for these equations.
The boundary conditions for velocity and temperature are clear. They are:y0;yH;uuooTTTHTC.The boundary conditions on species are not so simple. We have to determineY Si H 4 , YS i H , and YH at y = 0 and H. Now, SiH 2 is an unsaturated molecule,22so we assume that each molecule that strikes a surface reacts with unit probability. In that case, the proper boundary condition is22Chemical Vapor Deposition for MicroelectronicsYSiH2= 0 aty= 0, Hwhere each Si H 2 molecule leaves one Si atom on the surface and one H 2 molecule leaves the surface.The flux of SiH 4 molecules into a solid surface depends on whether theyare destroyed at the surface or reflected. If reflected, the net flux is zero.
Ifdestroyed, the net flux is a maximum. For SiH 4 , some are reflected and someare destroyed.The fraction of silane molecules that adsorb and decompose upon collisionwith a solid surface can be estimated from experimental data.Then the boundary condition on YSiH 4 can be derived by equating theflux as calculated from continuum arguments to the flux, as computed fromkinetic theory. The result is a mixed, nonlinear boundary condition involvingYSiH 4 and dYSiH4' From this, we can evaluate the rate at which a silicon filmwill grow on the hot wall.The boundary condition on V H 2 can be determined if we remember thateach Si H 2 and each Si H 4 molecule releases H 2 as it decomposes on the surface.Then we can write(41 )eval uated at the wall.Finally, we require expressions for J.1, k, and Dj as functions of T and Vj'sbefore we can solve our equations.
As noted earlier, they can be derived from1ski netic theory, and an explanation of how they are developed is available.Equations (30), (31), and (32) are all highly nonlinear differential equations, so we will solve them by replacing derivatives with finite differences anduse a high-speed digital computer to solve the resulting difference equations.Before discussing solution techniques, it is interesting to make the following observations:(1) The momentum equation depends only on T through the temperature dependency of J.1.(2) The energy equation requires a knowledge of the V's, but is independent of u.(3) The species conservation equations depend on T, but are also independent of u.Therefore, we can solve the energy and species equations to obtain values forthe ViS and T, and then use these to calculate u.The boundary conditions for the solution are:u(o)T(o)= u(L) = 0= TH, T(L) = Teand the conditions on the V's discussed earlier.Fundamentals of Thermal CVD23The momentum and energy equations are solved using a point-by-pointiteration scheme.
Derivatives are first replaced by finite differencies. A typicalpoint is shown belowyNpSand we write, for any functions, ¢ and t/JandThen, Equations (3D) and (31) are written asMomentum:~(42)dxEnergy:(43)Lnj =1!PPVj(V ) (c p )P Yj Pj P[Tfj- T ]S2h+ W.w· (cp.l p TpJP JJIn the energy equation, we can replace (V y j)p from Equation (33) so thatEquation (45) can be rewritten as:24Chemical Vapor Deposition for Microelectronicsi=(44)j=lYpP~p\1-~Yjpc ) [TN - TS]2h( PJ0 Po_or, uSing wY j= WjX j+i==1wJowJoPJO(C p 0)JTpPpRT---=-,we getand p =Wp wpR+For a small degree of dissociation, we assumecan be simpl ified as(TN - T5 ) pw n-2~---J:-L-Rj=l()CPjwn = Ws = wp , and Equation (46)Dj P ( Yj N- Yj S)~1.+ LP Jpk=lYYo\jp0kp(Y-y)kN kSI_I- 0At a typical grid point, we assume we know TN and T s and wish to solve forT p.
If the iteration is proceeding upward (y positive), then Ts for the firstinterior point is known from the boundary condition and TN is known from theinitial guess. -rhe thermal conductivity, k, tt\e net production rates, Wj, and theFundamentals of Thermal CVD25diffusion coefficients, OJ, are calculated from the initial guess for T and the assumed known solution for the V's. We then solve the quadratic equation forT p at the first interior point. Next, the following point is considered and Tsfor it is the just-calculated T p from the first point.
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