Thermochemical Processes Episode 4

80 Thermochemical Processes: Principles and Models time is eliminated as a variable by using the Laplace transform L.x,t/ thus Z 1 L[.Q.x,t/] D e stQ.x,t/dt 0 ∂2Q d2Q ∂x2 dx2 L∂Q D sQ Tables of Laplace transforms for a large number of functions have been calcu-lated, and can be obtained from published data. In the present example, the transformed equation is dx2 sQ D 0 for which the solution is Q D A1e q0x C A2eq0x where q02 D s/˛. Applying the boundary conditions it follows that A2 above must be zero since Q remains finite as x ! 1. At x equal to zero, Q D Q0 thus A1 D Q0 From tables for the Laplace function the inverse Laplace function can then be read and this yields Q D Q0 erfc xp˛t which is given above for the solution to the laser heating problem. An alternative method of solution to these analytical procedures, which is particularly useful in computer-assisted calculations, is the finite-difference technique. The Fourier equation describes the accumulation of heat in a thin slice of the heated solid, between the values x0 and x0 C dx, resulting from the flow of heat through the solid. The accumulation of heat in the layer is the difference between the flux of energy into the layer at x D x0, Jx0 and the flux out of the layer at x D x0 C dx, Jx0Cdx. Therefore the accumulation of heat in the layer may be written as dx ∂t D Jx0 Jx0Cdx D dx ∂x xDx0 but at any given point in the heat profile in the solid J D ˛ ∂x hence ∂t D ∂x ˛ ∂x Gaseous reaction kinetics and molecular decomposition 81 which is the Fourier equation. For the numerical solution of this equation the variables are first changed to dimensionless variables q D Q/Q0; ˇ D x/l; D ˛t/l2 where l is the total thickness of the substrate. The heat conduction equation in terms of these variables has the components ∂Q ∂Q dˇ ∂Q 1 ∂x ∂ˇ dx ∂ˇ l ∂Q ∂Q d ∂Q ˛ ∂t ∂ dt ∂ l2 ∂2Q ∂2Q 1 ∂x2 ∂ˇ2 l2 The thickness of the solid is then divided into thin slices, and the separate differentials at the mth slice in the Fourier equation can be expressed in terms of the functions ∂2q! .qmC1 qm/ .qm qm 1/ ∂ˇ2 .∂ˇ/2 ∂q .qŁ qm/ ∂t m υ where qŁ is the value of q after a time increment υ, and finally, on substituting in the heat conduction equation, qŁ D qm C .υˇ/2 .qmC1 2qm C qm 1/ Successive steps in this distribution can then be obtained to calculate the values of Q at a given value of x in a sequence of time intervals. Methods for numerical analyses such as this can be obtained from commer-cial software, and the advent of the computer has considerably eased the work required to obtain numerical values for heat distribution and profiles in a short time, or even continuously if a monitor supplies the boundary values of heat content or temperature during an operation. Returning to laser heating of the film which is deposited on a substrate, it is possible to control the temperature of the film and the substrate through the power of the light source or by scanning the laser beam across the surface of the substrate at a speed which will allow enough time for the deposit to be formed (laser writing). In the practical situation it is simpler to move the substrate relative to the laser beam in order to achieve this. The speed must be selected so as to optimize the rate of formation of the film while maintaining the desired relation of the physical properties of the film and the substrate such as cohesion and epitaxial growth. 82 Thermochemical Processes: Principles and Models Radiation and convection cooling of the substrate This calculation is subject to two further considerations. The first of these is that a substrate such as quartz, which is transparent in the visible region, will not absorb all of the incident light transmitted through the gas phase. The Fourier calculation shown above considers only the absorbed fraction of the energy at the surface. Secondly, if the absorption of radiation at the surface of the substrate is complete, leading to the formation of a hot spot, this surface will lose heat to its cooler surroundings by radiation loss. The magnitude of this loss can be assessed using the Stefan–Boltzmann law, Qx D Fe.T4 T4 / Here, Qr is the energy loss per second by a surface at temperature Ts to its surroundings at temperature Tm, the emissivity of the substrate being e, the view factor F being the fraction of the emitted radiation which is absorbed by the cool surroundings, and being the Stefan–Boltzmann radiation constant (5.67 ð 10 8 Jm 2 s 1 K 4). In the present case, the emissivity will have a value of about 0.2–0.3 for the metallic substrates, but nearly unity for the non-metals. The view factor can be assumed to have a value of unity in the normal situation where the hot substrate is enclosed in a cooled container. There will also be heat loss from the substrate due to convection currents caused by the temperature differential in the surrounding gas phase, but this will usually be less than the radiation loss, because of the low value of the heat transfer coefficient, h, of gases. The heat loss by this mechanism, Qc, can be calculated, approximately, by using the Richardson–Coulson equation Qc D h.Ts Tm/ D 5.6.Ts Tm/1/4 Jm 2 s 1 K 1 which indicates a heat loss by this mechanism which is about one quarter of the radiation loss for a substrate at 1000K and a wall temperature of 300K. The major effect of the convection currents will be to mix the gas phase so that the surface of the substrate does not become surrounded by the reaction products. Laser production of thin films There are therefore two ways in which lasers may be used to bring about photon-assisted film formation. If the laser emits radiation in the near-ultra-violet or above, photochemical decomposition occurs in the gas phase and some unabsorbed radiation arrives at the substrate, but this latter should be a minor effect in the thin film formation. This procedure is referred to as photol-ysis. Alternatively, if the laser emits radiation in the infra-red, and the photons are only feebly absorbed to raise the rotational energy levels of the gaseous Gaseous reaction kinetics and molecular decomposition 83 species, the major absorption occurs at the substrate, which therefore generates a ‘hot-spot’, as described above, at the point where the beam impinges. This spot heats the gas phase in its immediate environment to bring about thermal dissociation. This is therefore a pyrolytic process. Lasers which provide a continuous source of infra-red are of much greater power than the pulsed sources which operate at higher frequencies, so the pyrolytic process is more amenable to industrial processing where a specific photochemical reaction is not required. Because of the possibility of focusing laser beams, thin films can be produced at precisely defined locations. Using a microscope train of lenses to focus a laser beam makes possible the production of microregions suitable for application in computer chip production. The photolytic process produces islands of product nuclei, which act as preferential nucleation sites for further deposition, and thus to some unevenness in the product film. This is because the substrate is relatively cool, and therefore the surface mobility of the deposited atoms is low. In pyrolytic decomposition, the region over which deposition occurs depends on the thermal conductivity of the substrate, being wider the lower the thermal conductivity. For example, the surface area of a deposit of silicon on silicon is narrower than the deposition of silicon on silica, or on a surface-oxidized silicon sample, using the same beam geometry. The energy densities of laser beams which are conventionally used in the production of thin films is about 103 104 Jcm 2 s 1, and a typical substrate in the semiconductor industry is a material having a low thermal conductivity, and therefore the radiation which is absorbed by the substrate is retained near to the surface. Table 2.8 shows the relevant physical properties of some typical substrate materials, which can be used in the solution of Fourier’s equation given above as a first approximation to the real situation. Table 2.8 Thermal conductivities and heat capacities of some metals and oxides Material Thermal conductivity (Wm 1 K 1) Heat capacity (Jmol 1 K 1) 300K 1300K Copper 397 244 Iron 73.3 28 Silicon 138 Beryllia 202 15 Magnesia 46 6.3 Silica 1.5 2.5 Alumina 39 5 300K 1300K 24.35 32.19 25.02 34.84 20.0 26.57 25.38 51.58 37.43 52.78 44.82 71.96 79.36 128.78 84 Thermochemical Processes: Principles and Models Molecular decomposition in plasma systems Apart from thermal and photon decomposition, the production of atoms and radicals in gaseous systems always occurs in plasma. This is because of the presence of both high energy electrons as well as photons in the plasma volume. The energy spectrum of the electrons is determined by the ioniza-tion potential of the plasma gas, the mean free path of the electrons between collisions, which depends on the pressure, and the applied electric poten-tial gradient. Measurements of the energy spectrum in a glow discharge have yielded electron energy values of the order of 2–10eV (200–1000kJmole 1), and the photon spectrum can extend to 40eV. The plasma therefore contains electrons and photons which can produce the free radicals which initiate chain reactions, and the novel feature is the capability to produce significant quanti-ties of ionized gaseous species. The ions which are co-produced in the plasma have a much lower temperature, around 7–800K, and hence the possibility that the target will be excessively heated during decomposition using plasma is very unlikely. However, there is not the capability of easy control of the energy of the dissociating species as may be exercised in photodecomposition. All of the atomic species which may be produced by photon decomposition are present in plasma as well as the ionized states. The number of possible reactions is therefore also increased. As an example, the plasma decomposition of silane, SiH4, leads to the formation of the species, SiH3, SiH2, H, SiHC, SiH3C and H2C. Recombination reactions may occur between the ionized states and electrons to produce dissociated molecules either directly, or through the intermediate formation of excited state molecules. AB C e ! A C B : AB C e ! ABŁ C e ! A C B C e where ABŁ represents a molecule in an excited state in which an energy level is reached involving some electron re-arrangement, such as spin decoupling in the stable bonding configuration of the molecule. The lifetime of these excited states is usually very short, of the order of 10 7 seconds, and thus they do not play a significant part in reactions which normally occur through the reaction of molecules or atoms in the ground, most stable, state. They may provide the activation energy for a reaction by collision with normal molecules before returning to the ground state, similar to the behaviour of the activated molecules in first-order reactions. A useful application of plasma is in the nitriding of metals or the formation of nitrides. Thermal methods for this require very high temperatures using nitrogen gas as the source, due to the high stability of the nitrogen molecule, and usually the reaction is carried out with ammonia, which produces nitrogen and hydrogen by dissociation. There is therefore a risk of formation of a nitro-hydride in some metals, such as titanium and zirconium, which form stable hydrides as well as the nitrides. In a nitrogen plasma, a considerable degree of ... - tailieumienphi.vn