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80 Thermochemical Processes: Principles and Models time is eliminated as a variable by using the Laplace transform Lx, t thus L[Qx, t] D 1 0 e st Qx, t dt L ∂ 2 Q ∂x 2 D d 2 Q dx 2 ; L ∂Q ∂t D s Q 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 d 2 Q dx 2 s ˛ Q D 0 for which the solution is Q D A 1 e q 0 x C A 2 e q 0 x where q 02 D s/˛. Applying the boundary conditions it follows that A 2 above must be zero since Q remains finite as x !1.Atx equal to zero, Q D Q 0 s thus A 1 D Q 0 s From tables for the Laplace function the inverse Laplace function can then be read and this yields Q D Q 0 erfc x 2 p ˛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 x 0 and x 0 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 x 0 , J x 0 and the flux out of the layer at x D x 0 C dx, J x 0 Cdx . Therefore the accumulation of heat in the layer may be written as dx ∂Q ∂t D J x 0 J x 0 Cdx Ddx ∂J ∂x xDx 0 but at any given point in the heat profile in the solid J D ˛ ∂Q ∂x hence ∂Q ∂t D ∂ ∂x ˛ ∂Q ∂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/Q 0 ; ˇ D x/l; D ˛t/l 2 where l is the total thickness of the substrate. The heat conduction equation in terms of these variables has the components ∂Q ∂x D ∂Q ∂ˇ dˇ dx D ∂Q ∂ˇ 1 l ; ∂ 2 Q ∂x 2 D ∂ 2 Q ∂ˇ 2 1 l 2 ∂Q ∂t D ∂Q ∂ d dt D ∂Q ∂ ˛ l 2 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 ∂ 2 q ∂ˇ 2 m D q mC1 q m q m q m1 ∂ˇ 2 ∂q ∂t m D q Ł m q m υ where q Ł is the value of q after a time increment υ, and finally, on substituting in the heat conduction equation, q Ł m D q m C υ υˇ 2 q mC1 2q m C q m1 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, Q x D FeT 4 s T 4 m Here, Q r is the energy loss per second by a surface at temperature T s to its surroundings at temperature T m , 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 ð10 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, Q c ,can be calculated, approximately, by using the Richardson–Coulson equation Q c D hT s T m D 5.6T s T m 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 1000 K and a wall temperature of 300 K. 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 10 3 10 4 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. Tab le 2.8 Thermal conductivities and heat capacities of some metals and oxides Material Thermal conductivity Heat capacity (W m 1 K 1 )(Jmol 1 K 1 ) 300 K 1300 K 300 K 1300 K Copper 397 244 24.35 32.19 Iron 73.3 28 25.02 34.84 Silicon 138 20.0 26.57 Beryllia 202 15 25.38 51.58 Magnesia 46 6.3 37.43 52.78 Silica 1.5 2.5 44.82 71.96 Alumina 39 5 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 2–10 eV (200–1000 kJ mole 1 ), and the photon spectrum can extend to 40 eV. 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 7–800 K, 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, SiH 4 , leads to the formation of the species, SiH 3 ,SiH 2 ,H,SiH C 2 , SiH 3 C and H 2 C . 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 Ce ! A C B:ABC e ! AB Ł C e ! A C B C e where AB Ł 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 Gaseous reaction kinetics and molecular decomposition 85 dissociation of the nitrogen molecules occurs, together with positively charged species such as N C and N 2C . The rate of nitriding of metals such as titanium may be increased by imposing a negative potential on the metal to be nitrided with respect to the plasma. Carbides may also be prepared, either by direct carburizing, as in the case of steel, in which a surface carbide film dissolves into the substrate steel, or by refractory metal carbide formation as in the cases when one of the refractory metal halides is mixed with methane in the plasma gas. Bibliography D.A. Eastham. Atomic Physics of Lasers, Taylor & Francis, London (1986) QC 688 E37. J. Mazumdar and A. Kar. Laser Chemical Vapor Deposition, Plenum NY (1995) TS 695 M39. J.G. Eden. Photochemical Vapor Deposition, J. Wiley (1992) TS 693 E33. B. Koplitz, Z. Xu and C. Wittig, Appl. Phys. Lett., 52, 860 (1988). K. Kamisako, T. Imai, and Y. Tarui, Jpn. J. Appl. Phys., 27, 1092 (1988). L. Hellner, K.T.V. Grattan and M.H.R. Hutchinson, J. Chem. Phys., 81, 4389 (1984). V.G. Jenson and G.V. Jeffreys. Mathematical Methods in Chemical Engineering, Academic Press, London (1963). M. Venugopalan and R. Avni, Chapter 3 in Klabunde loc. cit. (1985). Chapter 3 Vapour phase transport processes Vapour transport processes A rapidly developing technique in the materials science of thin film and single crystal growth involves the transport of atoms across a temperature gradient by means of gas–solid reactions. The objectives may be many, and include the separation of elements from an unwanted impurity and the removal of atoms from a polycrystalline sample to form a single crystal among others. The atomic species are transported by molecules which are formed by reaction of the solid with a specially chosen reactant gas, and may occur up or down the temperature gradient, depending on the nature of the transport reaction. In most circumstances, it can be assumed that the gas–solid reaction pro- ceeds more rapidly than the gaseous transport, and therefore that local equi- librium exists between the solid and gaseous components at the source and sink. This implies that the extent and direction of the transport reaction at each end of the temperature gradient may be assessed solely from thermodynamic data, and that the rate of transport across the interface between the gas and the solid phases, at both reactant and product sites, is not rate-determining. Transport of the gaseous species between the source of atoms and the sink where deposition takes place is the rate-determining process. Thermodynamics and the optimization of vapour phase transport The choice of the transporting reagent for a given material is made so that the reaction is as complete as possible in one direction, in the uptake, and the reverse reaction in the opposite direction at the deposition site. This requires that not only the choice of the reagent, but also the pressure and temper- ature ranges under which the reaction is most effectively, or quantitatively, performed, must be calculated (Alcock and Jeffes, 1967; 1968). There will always be limitations placed on this choice by the demands of the chemical inertness and temperature stability of the containing materials in which the reaction is carried out. These considerations apart, the selection of the optimum conditions for the performance of a transporting reaction requires the choice of the best average value of the equilibrium constant. The effect of the range in the Vapour phase transport processes 87 Tab le 3.1 Val ues of pXY 2 at equilibrium with Xs at varying values of Kp D 1 atmos for the reaction X(s) CY 2 (g) D XY 2 (g) KpXY 2 (atmos) 10 3 9.1 ð10 4 10 2 9.1 ð10 3 10 1 9.1 ð10 2 10.50 10 0.91 10 2 0.99 10 3 0.999 equilibrium constant of a transporting reaction on the partial pressure variation of a transporting gas is exemplified by Table 3.1 for the reaction: X(s) CY 2 (g) D XY 2 (g) This shows the partial pressure of transporting molecules which are formed at various values of the equilibrium constant, for the reaction carried out at one atmosphere pressure. It can be seen that the greatest change in the equilibrium partial pressure occurs around the value of the equilibrium constant of unity, and hence where the standard Gibbs energy change has the value zero. It is true that the greatest value of the partial pressure of the transporting molecule is to be found at the highest value of the equilibrium constant, but the greatest change is about the value of unity. The optimal choice depends on the total pressure of the system, and on the stoichiometry of the reaction. As an example, the transportation of zirconium as the tetra-iodide is made at low pressure, while the purification of nickel by tetracarbonyl formation is made at high pressure. These reactions may be written as Zr C4I(g) D ZrI 4 (g) G ° D784 900 C 321.73T Jmol 1 D 0 at 2440 K and Ni C4CO(g) D Ni(CO) 4 G ° D138 860 C 368.7T Jmol 1 D 0 at 377 K 88 Thermochemical Processes: Principles and Models Both reactions involve the formation of a vapour-transporting species from four gaseous reactant molecules, but whereas the tetra-iodide of zirconium is a stable molecule, the nickel tetracarbonyl has a relatively small stability. The equilibrium constants for these reactions are derived from the following considerations: For the general reaction X(s) C4Y(g) D XY 4 (g) when x moles of product are formed, leaving 41 x moles of reactant gas, and yielding a total number of moles 4 3x, the equilibrium constant is given by K D 1 P 3 x4 3x 3 4 4x 4 where P is the total pressure. Considering only the value of the equilibrium constant, and ignoring the temperature effect on the actual value in any partic- ular process, the optimum value of K will be found at the maximum value of the efficiency function F D p dp/dK for the product gas. Efficient transport of material across a temperature gradient depends not only on the change in the equilibrium constant of the transporting reaction, but also on the mean partial pressure of the transporting species. The product p dp/dK can be used to assess the effectiveness of a given reaction, and the maximum of this function is to be found at different values of K, depending on the total pressure and the stoichiometry. According to Le Chatelier’s principle it is to be expected that a change in total pressure will have an effect on the maximum of this function, and in the examples of zirconium and nickel refining given above, there is a marked difference in the optimum conditions for carrying out these reactions. This is due to the large difference in the heats of formation of these transporting species and the effect of these heats on the value of the equilibrium constants. Figure 3.1 shows that the optimum for a reaction involving no change in the number of gaseous molecules, which we shall call a 1:1 reaction, is independent of the total pressure. The change due to pressure in a 4:1 reaction, such as the refining reactions discussed above, is very marked however, leading to different values in the optimum value of the equilibrium constant, thus a low pressure system is best for zirconium transport where the equilibrium favours the iodide, while a high-pressure system is used for nickel purification, since the equilibrium constant produces a small partial pressure of the carbonyl at one atmosphere pressure. Vapour phase transport processes 89 The direction of vapour transport across a thermal gradient In some transport reactions the transporting species is carried up a temperature gradient and in others the transport is in the opposite direction. For example the transport of aluminium by reaction with AlCl 3 to form the monochloride, AlCl, occurs down the temperature gradient. This reaction is written as 2Al CAlCl 3 (g) D 3AlCl(g) G ° D 391 600 261.3T Jmol 1 and the lower chloride is the transporting gas. The difference between this reac- tion, a 1:3 reaction, and the zirconium transport reaction 4:1, which occurs up a temperature gradient, lies in the entropy change. This is mainly determined by the change in the number of gaseous molecules which occurs in each reac- tion. The 1:3 reaction shown above will be accompanied by an increase in the 10 6 10 4 10 2 10 −2 10 −4 10 −6 10 −8 10 −10 10 −12 10 −14 10 −16 1 −10 −8 −6 −4 −2024681012 ∑ P = 10 atm ∑ P = 1.0 atm ∑ P = 0.1 atm ∑ P = 0.01 atm X (s,l) + Y (g) = XY (g) P XY × d P XY /d K Log K (a) Figure 3.1 (a) & (b) The efficiency of 1:1 and 1:4 vapour phase transport reactions, showing the marked dependence of the optimum for the 1:4 reaction on the pressure. (c) The dependence of the 1:2 and 1:4 reaction Gibbs energy on temperature and pressure, showing that the formation of nickel carbonyl is favoured by high pressure, and that of zirconium tetra-iodide, which is much more stable, is favoured by low pressures [...]...90 Thermochemical Processes: Principles and Models Log K −10 −8 −6 −2 0 2 4 6 8 10 12 X(s,l) + 4Y(g) = XY4(g) 0 at m 1 04 4 ∑ = 10 102 1 P m PXY4 × dPXY4 / dK 10−2 10 4 ∑P = 0 1 at 10−6 m 10−8 ∑P = 1 0 at 10−10 m 10−12 ∑P 10− 14 = 1 0 at 0 10−16 (b) 10−5 4) 40 0 −10 −20 −30 40 O i(C ∑P = CO +4 Ni ∆G °(kcal) 10 X(s,l) + 2Y(g) = XY2(g) +N 30 20 10 4 X(s,l) + 4Y(g) = XY4(g) 0 ∑P = 1 500... follows: TiB2 : TiCl4 (g) C 2BCl3 (g) C 5H2 ; G° D 298 780 260.6T TiBr4 (g) C 2BBr4 (g) C 5H2 ; G° D 255 150 265T G° D 67 140 258.8T TiI4 (g) C 2BI3 (g) C 5H2 ; ZrB2 : ZrCl4 (g) C 2BCl3 (g) C 5H2 ; G° D 40 2 760 G° D 336 100 253.1T ZrI4 (g) C 2BI3 (g) C 5H2 ; G° D 126 40 0 257.7T HfB2 : HfCl4 (g) C 2BCl3 (g) C 5H2 ; G° D 40 9 840 256.5T HfBr4 (g) C 2BBr3 (g) C 5H2 ; G° D 348 250 252.9T HfI4 (g) C 2BI3... Partial pressures of some species in the gaseous system Si–C–H–Cl at 1195 K Gaseous species CH4 SiCl4 H2 HCl C2 H4 C2 H2 C2 H6 SiH4 SiCl3 SiH4 Partial pressure (atmos) 3.51 ð 10 4. 60 ð 10 1.29 ð 10 2.52 ð 10 3.36 ð 10 2.09 ð 10 7.17 ð 10 7.35 ð 10 7. 04 ð 10 9.85 ð 10 1 1 1 3 2 2 4 4 4 5 98 Thermochemical Processes: Principles and Models It therefore appears that other hydrocarbons than methane take... iodides of these metals compared to that of zirconium can be calculated from the exchange reactions Zr C SiI4 (g) D Si C ZrI4 (g) : G° D 550 log K D log 8.26T D 89 040 150.4T D 246 500 aSipZrI4 (g) D 21.5 aZrpSiI4 (g) 3Zr C 2Al2 I6 (g) D 4Al C 3ZrI4 (g) : G° D log K D log 241 600 171 019 a4 AlpZrI4 (g) D 16.20 a3 Zrp2 Al2 I6 (g) and there are no reliable data for the iron gaseous iodides Since silicon... minimization calculation should be made 95 ∆m negative Vapour phase transport processes 0 Cl 4 6 Si 2 8 4 6 C 2 8 4 6 Cl 8 ∆m negative 2 0 Cl 2 4 6 8 Si 2 4 6 8 C 2 4 6 8 H 2 4 6 8 Cl Figure 3.2 Chemical potential diagrams for the transport of silicon carbide by chlorine, showing that the much greater stability of SiCl4 than CCl4 makes this process very inefficient, while the use of HCl as the transporting gas... of the halides Thus H° SiCl4 H° TaCl4 D 662.7 566.9 D 95.8 kJ mol 1 The heat of formation of SiI4 (g) is 125.1 kJ mol 1 and so the heat of formation of TaI4 may be estimated to be 29 kJ mol 1 , and the entropies may be estimated from the empirical equation S° 298 D 131.8 C 207.1 log M (Kubaschewski et al loc cit.) to be for SiI4 , S° 298 D 43 3 and for TaI4 , S° 298 D 45 6 J K This estimated Gibbs... 298 D 45 6 J K This estimated Gibbs energy of the transporting reaction is, Ta C Si C 4I2 (g) D TaI4 (g) C SiI4 (g) D 1 mol 1 396 290 C 220.4T which passes through zero at 1800 K The heats of formation of the silicides Ta5 Si3 and Ta3 Si2 are about 40 kJ gram-atom 1 , with entropies of formation Vapour phase transport processes 99 approximately equal to zero entropy units per gram-atom The Gibbs energy... reaction to form the gaseous iodides from the compound Ta5 Si3 is therefore obtained from the reactions 5Ta C 3Si C 16I2 (g) D 5TaI4 (g) C 3SiI4 (g); G° D G° D 5Ta C 3Si D Ta5 Si3 ; 1 49 0 610 C 8 64. 8T J 339 300 C 9.8T J and hence Ta5 Si3 C 16I2 (g) D 5TaI4 (g) C 3SiI4 (g) G° D 1 151 300 C 873.8T J which passes through zero at approximately 1300 K The chemical potential diagram for this ternary system... C 4HCl(g) D SiCl4 (g) C CH4 (g) G° D 305 040 C 255.3T J mol 1 D 0 at 1195 K This is a 4: 2 reaction, and is thus pressure dependent However, it is necessary to compute the equilibrium partial pressure of some alternative gaseous species, such as SiCl3 , and other hydrocarbons such as C2 H2 and for this a Gibbs energy minimization calculation should be made 95 ∆m negative Vapour phase transport processes. .. trichloride for the transport of gallium after arsenic has evaporated as the element or as a chloride The probability 94 Thermochemical Processes: Principles and Models of formation of the monochloride may be determined from the data for the reaction GaCl3 (g) C 2Ga D 3GaCl(g) G° D 196 900 249 .4T J mol 1 D 0 at 790 K Below this temperature, mainly GaCl3 transports gallium, but above this temperature, the . 244 24. 35 32.19 Iron 73.3 28 25.02 34. 84 Silicon 138 20.0 26.57 Beryllia 202 15 25.38 51.58 Magnesia 46 6.3 37 .43 52.78 Silica 1.5 2.5 44 .82 71.96 Alumina 39 5 79.36 128.78 84 Thermochemical Processes: . (atmos) CH 4 3.51 ð10 1 SiCl 4 4.60 ð10 1 H 2 1.29 ð10 1 HCl 2.52 ð10 3 C 2 H 4 3.36 ð10 2 C 2 H 2 2.09 ð10 2 C 2 H 6 7.17 ð10 4 SiH 4 7.35 ð10 4 SiCl 3 7. 04 ð10 4 SiH 4 9.85 ð10 5 98 Thermochemical. reactions Zr CSiI 4 (g) D Si CZrI 4 (g) : G ° 550 D 241 600 8.26T D 246 500 log K D log aSipZrI 4 (g) aZrpSiI 4 (g) D 21.5 3Zr C2Al 2 I 6 (g) D 4Al C 3ZrI 4 (g) : G ° D89 040 150.4T D171 019 log
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