28 A Review of Materials Science WEIGHT PER CENT SILICON Ge ATOMIC PER GENT SILICON Si Figure 1-1 2. Ge-Si equilibrium phase diagram. (Reprinted with permission from M. Hansen, Constitution of Binary Alloys, McGraw-Hill, Inc. 1958). wise, these diagrams hold at atmospheric pressure, in which case the variance is reduced by 1. The Gibbs phase rule now states f = n + 1 - J/ or f = 3 - II/. Thus, at most three phases can coexist in equilibrium. To learn how to interpret binary phase diagrams, let us first consider the Ge-Si system shown in Fig. 1-12. Such a system is interesting because of the possibility of creating semiconductors having properties intermediate to those of Ge and Si. On the horizontal axis, the overall composition is indicated. Pure Ge and Si components are represented at the extreme left and right axes, respectively, and compositions of initial mixtures of Ge and Si are located in between. Such compositions are given in either weight or atomic percent. The following set of rules will enable a complete equilibrium phase analysis for an initial alloy composition X, heated to temperature To. 1. Draw a vertical line at composition X, . Any point on this line represents a state of this system at the temperature indicated on the left-hand scale. 2. The chemical compositions of the resulting phases depend on whether the point lies (a) in a one-phase field, (b) in a two-phase field, or (c) on a sloping or horizontal (isothermal) boundary between phase fields. a. For states within a single-phase field., i.e., L (liquid), S (solid), or a compound, the phase composition or chemical analysis is always the same as the initial composition. 1.5. Thermodynamics of Materials 29 b. In a two-phase region, i.e., L + S, CY + 0, etc., a horizontal tie line is first drawn through the state point extending from one end of the two-phase field to the other as shown in Fig. 1-12. On either side of the two-phase field are the indicated single-phase fields (L and S). The compositions of the two phases in question are given by projecting the ends of the tie line vertically down and reading off the values. For example, if Xo = 40 at% Si and To = 1200 "C, X, = 34 at% Si and X, = 67 at% Si. c. State points located on either a sloping or a horizontal boundary cannot be analyzed; phase analyses can only be made above or below the boundary lines according to rules a and b. Sloping boundaries are known as liquidus or solidus lines when L/L + S or L + S/S phase field combinations are respectively involved. Such lines also represent solu- bility limits and are, therefore, associated with the processes of solution or rejection of phases (precipitation) from solution. The horizontal isothermal boundaries indicate the existence of phase transformations involving three phases. The following common reactions occur at these critical isotherms, where CY, 0 and y are solid phases: 1. Eutectic: L + Q! + 0 2. Eutectoid: y + CY + /3 3. Peritectic: L + CY + y 3. The relative amount of phases present depends on whether the state point lies in (a) a one-phase field or (b) a two-phase field. a. Here the one phase in question is homogeneous and present exclusively. Therefore, the relative amount of this phase is 100%. b. In the two-phase field the lever rule must be applied to obtain the relative phase amounts. From Fig. 1-12, state Xo, To, and the corre- sponding tie line, the relative amounts of L and S phases are given by x 100, (1-19) Xo -X, x 100; %S = XS -Xo xs -x, xs -x, %L = where %L plus %S = 100. (Substitution gives %L = (67 - 40)/(67 - 34) x 100 = 81.8, and %S = (40 - 34)/(67 - 34) x 100 = 18.2.) Equation 1-19 represents a definition of the lever rule that essentially ensures conservation of mass in the system. The tie line and lever rule can be applied only in a two-phase region; they make no sense in a one-phase region. Such analyses do reveal information on phase compositions and amounts, yet they say nothing about the physical appearance or shape that phases actually take. Phase morphology is dependent on issues related to nucleation and growth. 30 A Review of Materials Science AI ATOMIC PER CENT SILICON SI Figure 1-1 3. M. Hansen, Consfitution of Binary Alloys, McGraw-Hill. Inc. 1958). AI-Si equilibrium phase diagram. (Reprinted with permission from Before leaving the Ge-Si system, note that L represents a broad liquid solution field where Ge and Si atoms mix in all proportions. Similarly, at lower temperatures, Ge and Si atoms mix randomly but on the atomic sites of a diamond cubic lattice to form a substitutional solid solution. The lens-shaped L -!- S region separating the single-phase liquid and solid fields occurs in many binary systems, including Cu-Ni, Ag-Au, Pt-Rh, Ti-W, and Al,O,-Cr,O,. A very common feature on binary phase diagrams is the eutectic isotherm. The AI-Si system shown in Fig. 1-13 is an example of a system undergoing a eutectic transformation at 577 "C. Alloy films containing about 1 at% Si are used to make contacts to silicon in integrated circuits. The insert in Fig. 1-13 indicates the solid-state reactions for this alloy involve either the formation of an Al-rich solid solution above 520 "C or the rejection of Si below this temperature in order to satisfy solubility requirements. Although this particu- 1.5. Thermodynamics of Materials 31 1000 I 800 ! i 600 1 lar alloy cannot undergo a eutectic transformation, all alloys containing more than 1.59 at% Si can. When crossing the critical isotherm from high tempera- ture, the reaction / . I / / / I - 577 "C L(11.3at% Si) A1(1.59at% Si) + Si ( 1-20) ! 400 200 29.5' n occurs. Three phases coexist at the eutectic temperature, and therefore f = 0. Any change in temperature and/or phase composition will drive this very special three-phase equilibrium into single- (i.e., L) or two-phase fields (i.e., L + Al, L + Si, A1 + Si), depending on composition and temperature. The important GaAs system shown in Fig. 1-14 contains two independent side-by-side eutectic reactions at 29.5 and 810 "C. For the purpose of analysis one can consider that there are two separate eutectic subsystems, Ga-GaAs and GaAs-As. In this way complex diagrams can be decomposed into simpler units. The critical eutectic compositions occur so close to either pure compo- nent that they cannot be resolved on the scale of this figure. The prominent central vertical line represents the stoichiometric GaAs compound, which melts at 1238 "C. Phase diagrams for several other important 3-5 semiconductors, 14001T 1200 WEIGHT PERCENT ARSENIC 810" "0 10 20 30 40 50 60 70 80 90 100 ATOMIC PERCENT ARSENIC AS Ga-As equilibrium phase diagram. (Reprinted with permission from Ga Figure 1-1 4. M. Hansen, Constitution of Binary Alloys, McGraw-Hill, Inc. 1958). 32 A Review of Materials Science (e.g., InP, GaP, and InAs) have very similar appearances. These compound semiconductors are common in other ways. For example, one of the compo- nents (e.g., Ga, In) has a low melting point coupled with a rather low vapor pressure, whereas the other component (e.g., As, P) has a higher melting point and a high vapor pressure. These properties complicate both bulk and thin-film single-crystal growth processes. We end this section on phase diagrams by reflecting on some distinctions in their applicability to bulk and thin-film materials. High-temperature phase diagrams were first determined in a systematic way for binary metal alloys. The traditional processing route for bulk metals generally involves melting at a high temperature followed by solidification and subsequent cooling to the ambient. It is a reasonable assumption that thermodynamic equilibrium is attained in these systems, especially at elevated temperatures. Atoms in metals have sufficient mobility to enable stable phases to nucleate and grow within reasonably short reaction times. This is not generally the case in metal oxide systems, however, because of the tendency of melts to form metastable glasses due to sluggish atomic motion. In contrast, thin films do not generally pass from a liquid phase through a vertical succession of phase fields. For the most part, thin-film science and technology is characterized by low-temperature processing where equilibrium is difficult to achieve. Depending on what is being deposited and the conditions of deposition, thin films possessing varying degrees of thermodynamic stability can be readily produced. For example, single-crystal silicon is the most stable form of this element below the melting point. Nevertheless, chemical vapor deposition of Si from chlorinated silanes at 1200 “C will yield single-crystal films, and amorphous films can be produced below 600 “C. In between, polycrystalline Si films of varying grain size can be deposited. Since films are laid down an atomic layer at a time, the thermal energy of individual atoms impinging on a massive cool substrate heat sink can be transferred to the latter at an extremely rapid rate. Deprived of energy, the atoms are relatively immobile. It is not surprising, therefore, that metastable and even amorphous semiconductor and alloy films can be evaporated or sputtered onto cool substrates. When such films are heated, they crystallize and revert to the more stable phases indicated by the phase diagram. Interesting issues related to binary phase diagrams arise with multicompo- nent thin films that are deposited in layered structures through sequential deposition from multiple sources. For example, ‘‘strained layer superlattices’ ’ of Ge-Si have been grown by molecular beam epitaxy (MBE) techniques (see Chapter 7). Films of Si and Si + Ge solid-solution alloy, typically tens of angstroms thick, have been sequentially deposited such that the resultant 1.6. Kinetics 33 composite film is a single crystal with strained lattice bonds. The resolution of distinct layers as revealed by the transmission electron micrograph of Fig. 14-17 is suggestive of a two-phase mixture. On the other hand, a single crystal implies a single phase even if it possesses a modulated chemical composition. Either way, the superlattice is not in thermodynamic equilibrium because the Ge-Si phase diagram unambiguously predicts a stable solid solution at low temperature. Equilibrium can be accelerated by heating, which results in film homogenization by interatomic diffusion. In thin films, phases such as solid solutions and compounds are frequently accessed horizontally across the phase diagram during an isothermal anneal. This should be contrasted with bulk materials, where equilibrium phase changes commonly proceed vertically downward from elevated temperatures. 1.6. KINETICS 1.6.1. Macroscopic Transport Whenever a material system is not in thermodynamic equilibrium, driving forces arise naturally to push it toward equilibrium. Such a situation can occur, for example, when the free energy of a microscopic system varies from point to point because of compositional inhomogeneities. The resulting atomic concentration gradients generate time-dependent, mass-transport effects that reduce free-energy variations in the system. Manifestations of such processes include phase transformations, recrystallization, compound growth, and degra- dation phenomena in both bulk and thin-film systems. In solids, mass transport is accomplished by diffusion, which may be defined as the migration of an atomic or molecular species within a given matrix under the influence of a concentration gradient. Fick established the phenomenological connection between concentration gradients and the resultant diffusional transport through the equation dC dx J= -D- (1-21) The minus sign occurs because the vectors representing the concentration gradient dC/& and atomic flux J are oppositely directed. Thus an increasing concentration in the positive x direction induces mass flow in the negative x direction, and vice versa. The units of C are typically atoms/cm3. The diffusion coefficient D, which has units of cm2/sec, is characteristic of both the diffusing species and the matrix in which transport occurs. The extent of 34 A Review of Materials Science observable diffusion effects depends on the magnitude of D. As we shall later note, D increases in exponential fashion with temperature according to a Maxwell-Boltzmann relation; Le., D = Doexp - E,/RT, ( 1-22) where Do is a constant and RT has the usual meaning. The activation energy for diffusion is ED (cal/mole) . Solid-state diffusion is generally a slow process, and concentration changes occur over long periods of time; the steady-state condition in which concentra- tions are time-independent rarely occurs in bulk solids. Therefore, during one-dimensional diffusion, the mass flux across plane x of area A exceeds that which flows across plane x + dx. Atoms will accumulate with time in the volume A dx, and this is expressed by dJ dJ dc ( dx ) dx dt JA- J+-dx A= Adx=-Adx. (1-23) Substituting Eq. 1-21 and assuming that D is a constant independent of C or x gives ac( X, t) a2c( X, t) =D at a x2 ( 1-24) The non-steady-state heat conduction equation is identical if temperature is substituted for C and the thermal diffusivity for D. Many solutions for both diffusion and heat conduction problems exist for media of varying geometries, constrained by assorted initial and boundary conditions. They can be found in the books by Carslaw and Jaeger, and by Crank, listed in the bibliography. Since complex solutions to Eq. 1-24 will be discussed on several occasions (e.g., in Chapters 8, 9, and 13), we introduce simpler applications here. Consider an initially pure thick film into which some solute diffuses from the surface. If the film dimensions are very large compared with the extent of diffusion, the situation can be physically modeled by the following conditions: C(x,O) = 0 at t = 0 ( 1 -25a) C(o0, t) = 0 at x = 03 for t > 0. (1-25b) The second boundary condition that must be specified has to do with the nature of the diffusant distribution maintained at the film surface x = 0. Two simple cases can be distinguished. In the first, a thick layer of diffusant provides an essentially limitless supply of atoms maintaining a constant surface concentra- tion Co for all time. In the second case, a very thin layer of diffusant provides an instantaneous source So of surface atoms per unit area. Here the surface for 03 > x > 0, 1.6. Kinetics 35 concentration diminishes with time as atoms diffuse into the underlying film. These two cases are respectively described by c(0, t) = c, lmc( x, t) dx = so Expressions for C( x, t) satisfying these conditions are respectively (1 -26a) ( 1 -26b) X C(x, t) = C0erfc- = c,, rn SO X2 c(x, t) = ~ exp - - 4Dt' (1-27b) and these represent the simplest mathematical solutions to the diffusion equa- tion. They have been employed to determine doping profiles and junction 1 10.' 10 10 -31 t\\ F-Gt \\GAUSSIAN j- Figure 1-15. Normalized Gaussian and Erfc curves of C/C, vs. x/m. Both logarithmic and linear scales are shown. (Reprinted with permission from John Wiley and Sons, from W. E. Beadle, J. C. C. Tsai, and R. D. Plummer, Quick Reference Manual for Silicon Integrated Circuit Technology, Copyright 0 1985, Bell Telephone Laboratories Inc. Published by J. Wiley and Sons). 36 A Review of Materials Science depths in semiconductors. The error function erf x/2a, defined by (1-28) is a tabulated function of only the upper limit or argument x/2fi. Normalized concentration profiles for the Gaussian and Erfc solutions are shown in Fig. 1-15. It is of interest to calculate how these distributions spread with time. For the erfc solution, the diffusion front at the arbitrary concentration of C(x, t)/C, = 1/2 moves parabolically with time as x = 2merfc-'(1/2) or x = 0.96m. When becomes large compared with the film dimensions, the assumption of an infinite matrix is not valid and the solutions do not strictly hold. The film properties may also change appreciably due to interdiffusion. To limit the latter and ensure the integrity of films, D should be kept small, which in effect means the maintenance of low temperatures. This subject will be discussed further in Chapter 8. 1.6.2. Atomistic Considerations Macroscopic changes in composition during diffusion are the result of the random motion of countless individual atoms unaware of the concentration gradient they have helped establish. On a microscopic level, it is sufficient to explain how atoms execute individual jumps from one lattice site to another, for through countless repetitions of unit jumps macroscopic changes occur. Consider Fig. 1-16a, showing neighboring lattice planes spaced a distance a, apart within a region where an atomic concentration gradient exists. If there are n, atoms per unit area of plane 1, then at plane 2, n2 = n, + (dn /dx) a,, a b. ooo 0 000 OE0 00 0 0-0 0 ouo Figure 1-16. Atomistic view of atom jumping into a neighboring vacancy. (a) Atomic diffusion fluxes between neighboring crystal planes. 1.6. Kinetics 37 where we have taken the liberty of assigning a continuum behavior at discrete planes. Each atom vibrates about its equilibrium position with a characteristic lattice frequency v, typically lOI3 sec -’. Very few vibrational cycles have sufficient amplitude to cause the atom to actually jump into an adjoining lattice position, thus executing a direct atomic interchange. This process would be greatly encouraged, however, if there were neighboring vacant sites. The fraction of vacant lattice sites was previously given by eCEflkT (see Eq. 1-3). In addition, the diffusing atom must acquire sufficient energy to push the surrounding atoms apart so that it can squeeze past and land in the so-called activated state shown in Fig. 1-16b. This step is the precursor to the downhill jump of the atom into the vacancy. The number of times per second that an atom successfully reaches the activated state is ve-‘JIkT, where ci is the vacancy jump or migration energy per atom. Here the Boltzmann factor may be interpreted as the fraction of all sites in the crystal that have an activated state configuration. The atom fluxes from plane 1 to 2 and from plane 2 to 1 are then, respectively, given as J,,, = -vexp 1 - -exp Ef - -(Ca,), ‘i 6 kT kT ( 1 -29a) 1 &f ‘i dC 6 kT kT J2+1 = -vexp - -exp - - where we have substituted Ca, for n and used the factor of 116 to account for bidirectional jumping in each of the three coordinate directions. The net flux JN is the difference or 1 ‘f JN = - -aivexp - -exp - 6 kT ( 1-30) By association with Fick’s law, D can be expressed as D = D,exp - ED/RT (1-31) with Do = (1/6)aiv and ED = (E~ + €/.)NA, where NA is Avogadro’s num- ber. Although the above model is intended for atomistic diffusion in the bulk lattice, a similar expression for D would hold for transport through grain boundaries or along surfaces and interfaces of films. At such nonlattice sites, energies for defect formation and motion are expected to be less, leading to higher diffusivities. Dominating microscopic mass transport is the Boltzmann factor exp - E,/RT, which is ubiquitous when describing the temperature dependence of the rate of many processes in thin films. In such cases the kinetics can be described graphically by an Arrhenius plot in which the [...]... of this chapter will be woven into the subsequent fabric of the discussion on the preparation and properties of thin films 1 An FCC film is deposited on the (100) plane of a single-crystal FCC substrate It is determined that the angle between the [lo01 directions in the film and substrate is 63.4" What are the Miller indices of the plane lying in the film surface? 44 A Review of Materials Science 2. .. all areas of scientific research and technological endeavor The topics treated in this chapter will, therefore, deal with: 2. 1 Kinetic Theory of Gases 2. 2 Gas Transport and Pumping 2. 3 Vacuum Pumps and Systems 2. 1 KINETICTHEORY GASES OF 2. 1 I.Molecular Velocities The well-known kinetic theory of gases provides us with an atomistic picture of the state of affairs in a confined gas (Refs 1, 2) A fundamental... this case by the annular region between the concentric pipes 2. 2.3 Pumping Speed Pumping is the process of removing gas molecules from the system through the action of pumps The pumping speed S is defined as the volume of gas passing the plane of the inlet port per unit time when the pressure at the pump inlet is P Thus, S = Q/P (2- 16) Although the throughput Q can be measured at any plane in the system,... However, the kinetic energy of any collection of gas molecules is solely dependent on temperature For a mole quantity it is given by (1/ 2 ) M 7 = (3 /2) R T with (1 /2) R T partitioned in each of the coordinate directions 2. 1 .2 Pressure Momentum transfer from the gas molecules to the container walls gives rise to the forces that sustain the pressure in the system Kinetic theory shows that the gas pressure... conductance of aperture of 10 cm diameter = 11.7A = 1 1 7 ~ ( 5=~ ) 919L/sec C, = conductance of pipe 3 cm long = 1 2 2 D 3 / L = 12. 2(10)3/3 = 4065 L/sec C, = conductance of annular aperture = 11.7Aa,,,, = 11.7(0 .25 )a(1 02- g 2 ) = 331 L/sec = conductance of annular pipe = 12. 2 C, = C, = 4065 L/sec conductance of averture in end of Dive/diffusion DumD = 11.7- A C, C, (D, D , ) 2 ( D ,+ 0,) 12. 2(10 - 8)'(10... Review of Materials Science a FREE ENERGY APPLIED FIELD Figure 1-17 (a) Free-energy variation with atomic distance in the absence of an applied field (b) Free-energy variation with atomic distance in the presence of an applied field logarithm of the rate is plotted on the ordinate and the reciprocal of the absolute temperature is plotted along the abscissa The slope of the resulting line is then equal... distinguished from the previously discussed molecular collisions in the gas phase The number of molecules that strike an element of surface (perpendicular to a coordinate direction) per unit time and area is given by m v,dn, (P= Upon substitution of Eq 2- 2, we get n m 2 RT dv, = n i z (2- 7) The use of Eq 2- 3b yields (P = (1/4)nF, and substitution of the perfect gas law (Eq 2- 4) converts into the more recognizable... flow K n < 1, (2- 12a) intermediate flow 1 < Kn < 110, (2- 12b) Kn > 110 (2- 12c) viscous flow In air these limits can be alternatively expressed by DpP < 5 x cm-torr for molecular flow, and DpP > 5 x lo-' cm-torr in the case of viscous flow through the use of Eq 2- 5 Figure 2- 3 serves to map the dominant flow regimes on this basis Note that flow mechanisms may differ in various parts of the same system... assumption 49 50 Vacuum Science and Technology is that the large number of atoms or molecules of the gas are in a continuous state of random motion, which is intimately dependent on the temperature of the gas During their motion the gas particles collide with each other as well as with the walls of the confining vessel Just how many molecule-molecule or molecule-wall impacts occur depends on the concentration... , is the concentration of reactants at coordinate position 1 and C, the concentration of products at 2, then the net rate of reaction is proportional to r, = C,exp( - g) - C,exp( - '*iTAG), (1-36) 40 A Review of Materials Science t FREE ENERGY 1 - REACTION COORDINATE 2 Figure 1-18 Free-energy path for thermodynamically favored reaction 1 t 2 where G* is the free energy of activation As before, the . one end of the two-phase field to the other as shown in Fig. 1- 12. On either side of the two-phase field are the indicated single-phase fields (L and S). The compositions of the two. in the presence of an applied field. logarithm of the rate is plotted on the ordinate and the reciprocal of the absolute temperature is plotted along the abscissa. The slope of the resulting. which has units of cm2/sec, is characteristic of both the diffusing species and the matrix in which transport occurs. The extent of 34 A Review of Materials Science observable diffusion