The Materials Science of Thin Films 2011 Part 2 ppsx

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 bet

28 A Review of Materials Science

WEIGHT PER CENT SILICON

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 T o

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 C Y , 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 X o , T o , and the corre- sponding tie line, the relative amounts of L and S phases are given by

30 A Review of Materials Science

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-

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,

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

The minus sign occurs because the vectors representing the concentration

gradient d C / & 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 , / R T , ( 1-22) where Do is a constant and R T 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

( dx ) dx dt

J A - J + - d x A = - - A d x = - A d x (1-23) Substituting Eq 1-21 and assuming that D is a constant independent of C or x

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 ,

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 / 2 a , defined by

(1-28)

is a tabulated function of only the upper limit or argument x / 2 f i 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 =

2 m e r f c - ' ( 1 / 2 ) or x = 0 9 6 m 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 / d x ) a,,

Atomistic view of atom jumping into a neighboring vacancy

(a) Atomic diffusion fluxes between neighboring crystal planes

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 - - ( C a , ) , ‘i

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 , / R T , 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

38 A Review of Materials Science

a

APPLIED FIELD FREE

ENERGY

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 to - ED / R , from which the characteristic activation energy

can be extracted

The discussion to this point is applicable to motion of both impurity and matrix atoms In the latter case we speak of self-diffusion For matrix atoms there are driving forces other than concentration gradients that often result in transport of matter Examples are forces due to stress fields, electric fields, and interfacial energy gradients To visualize their effect, consider neighboring atomic positions in a crystalline solid where no fields are applied The free energy of the system has the periodicity of the lattice and varies schematically,

as shown in Fig 1-17a Imposition of an external field now biases the system such that the free energy is lower in site 2 relative to 1 by an amount 2 AG A free-energy gradient exists in the system that lowers the energy barrier to motion from 1 -+ 2 and raises it from 2 -+ 1 The rate at which atoms move from 1 to 2 is given by

sec-I

GD - AG

( 1 -32a)

1.6 Kinetics 39

Similarly,

r21 = vexp ( - G D i T A G ) sec-I, (1-32b) and the net rate r, is given by the difference or

= 2vexp - -si&- (1-33)

When AG = 0, the system is in thermodynamic equilibrium and r, = 0, so

no net atomic motion occurs Although GD is typically a few electron volts or

so per atom (1 eV = 23,060 cal/mole), AG is much smaller in magnitude

since it is virtually impossible to impose external forces on solids comparable

to the interatomic forces In fact, AGIRT is usually much less than unity, so

sinh AGIRT = A G I R T This leads to commonly observed linear diffusion effects But when AGIRT I :1, nonlinear diffusion effects are possible By multiplying both sides of Eq 1-33 by a o , we obtain the atomic velocity u :

2 AG

u = a,r, = [ a ; v e - ' ~ / ~ ~ ] -

a , R T ' ( 1-34) The term in brackets is essentially the diffusivity D with GD a diffusional activation energy (The distinction between G D and ED need not concern us here.) The term 2 A G l a , is a measure of the molar free-energy gradient or applied force F Therefore, the celebrated Nernst-Einstein equation results:

u = D F / R T (1-35) Application of this equation will be made subsequently to various thin-film mass transport phenomena, e.g , electric-field-induced atomic migration (elec- tromigration), stress relaxation, and grain growth The drift of charge carriers

in semiconductors under an applied field can also be modeled by Eq 1-35 In

some instances, larger generalized forces can be applied to thin films relative

to bulk materials because of the small dimensions involved

Chemical reaction rate theory provides a common application of the preced- ing ideas In Fig 1-18 the reactants at the left are envisioned to proceed toward the right following the reaction coordinate path Along the way, intermediate activated states are accessed by surmounting the free-energy

barrier Through decomposition of the activated species, products form If C ,

is the concentration of reactants at coordinate position 1 and C , the concentra-

tion 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 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 Boltzmann factors

represent the probabilities of surmounting the respective energy barriers faced

by reactants proceeding in the forward direction, or products in the reverse direction When chemical equilibrium prevails, the competing rates are equal and r, = 0 Therefore,

For the reaction to proceed to the right AG = GR - G p must be positive

By comparison with Eq 1-12, it is apparent that the left-hand side is the equilibrium constant and AG may be associated with - AGO This expression

is perfectly general, however, and applies, for example, to electron energy-level populations in semiconductors and lasers, as well as magnetic moment distribu- tions in solids In fact, whenever thermal energy is a source of activation energy, Eq 1-37 is valid

1.7 NUCLEATION

When the critical lines separating stable phase fields on equilibrium phase diagrams are crossed, new phases appear Most frequently, a decrease in

1.7 Nucleation 41

temperature is involved, and this may, for example, trigger solidification or solid-state phase transformations from now unstable melts or solid matrices When such a transformation occurs, a new phase of generally different structure and composition emerges from the prior parent phase or phases The process known as nucleation occurs during the very early stages of phase change It is important in thin films because the grain structure that ultimately develops in a given deposition process is usually strongly influenced by what happens during film nucleation and subsequent growth

Simple models of nucleation are first of all concerned with thermodynamic questions of the energetics of the process of forming a single stable nucleus Once nucleation is possible, it is usual to try to specify how many such stable nuclei will form within the system per unit volume and per unit time-i.e., nucleation rate As an example, consider the homogeneous nucleation of a

spherical solid phase of radius r from a prior supersaturated vapor Pure

homogeneous nucleation is rare but easy to model since it occurs without benefit of complex heterogeneous sites such as exist on an accommodating substrate surface In such a process the gas-to-solid transformation results in a

reduction of the chemical free energy of the system given by (4/3)7rr3AGv, where AG, corresponds to the change in chemical free energy per unit

volume For the condensation reaction vapor (v) + solid (s), Eq 1-13 indi-

cates that

k T P- k T P

(1-38)

where P, is the vapor pressure above the solid, P, is the pressure of the

supersaturated vapor, and Q is the atomic volume A more instructive way to

write Eq 1-38 is

AG, = - ( k T / Q ) l n ( l + S ) , ( 1-39)

where S is the vapor supersaturation defined by (P, - P,)/ P, Without

supersaturation, AGv is zero and nucleation is impossible In our example,

however, P, > P, and AGv is negative, which is consistent with the notion of

energy reduction Simultaneously, new surfaces and interfaces form This results in an increase in the surface free energy of the system given by 47rr2y,

where y is the surface energy per unit area The total free-energy change in forming the nucleus is thus given by

and minimization of AG with respect to r yields the equilibrium size of

r = r* Thus, d A G / d r = 0, and r* = - 2 y / A G v Substitution in Eq 1-40

42 A Review of Materials Science

gives AG* = 1 6 ~ y ~ / 3 ( A G , ) ~ The quantities r* and AG* are shown in Fig 1-19, where it is evident that AG* represents an energy barrier to the

nucleation process If a solid-like spherical cluster of atoms momentarily forms

by some thermodynamic fluctuation, but with radius less than r*, the cluster is

unstable and will shrink by losing atoms Clusters larger than r* have sur- mounted the nucleation energy barrier and are stable They tend to grow larger while lowering the energy of the system

The nucleation rate N is essentially proportional to the product of three terms, namely,

N = N * A * ~ (nuclei/cm2-sec) (1-41)

N* is the equilibrium concentration (per cm2) of stable nuclei, and w is the rate at which atoms impinge (per cm2-sec) onto the nuclei of critical area A* Based on previous experience of associating the probable concentration of an entity with its characteristic energy through a Boltzmann factor, it is appropri- ate to take N* = n,e-AG*/kT, where n, is the density of all possible nuclea- tion sites The atom impingement flux is equal to the product of the concentra- tion of vapor atoms and the velocity with which they strike the nucleus In the

next chapter we show that this flux is given by a ( P , - P,)N, I-,

Exercises 43

where A4 is the atomic weight and CY is the sticking coefficient The nucleus area is simply 4ar2, since gas atoms impinge over the entire spherical surface Upon combining terms, we obtain

The most influential term in this expression is the exponential factor It

contains AG*, which is, in turn, ultimately a function of S When the vapor supersaturation is sufficiently large, homogeneous nucleation in the gas is possible This phenomenon causes one of the more troublesome problems associated with chemical vapor deposition processes since the solid particles that nucleate settle on and are incorporated into growing films destroying their integrity

Heterogeneous nucleation of films is a more complicated subject in view of the added interactions between deposit and substrate The nucleation sites in this case are kinks, ledges, dislocations, etc., which serve to stabilize nuclei of differing size The preceding capillarity theory will be used again in Chapter 5

to model heterogeneous nucleation processes Suffice it to say that when ~ is high during deposition, many crystallites will nucleate and a fine-grained film results On the other hand, if nucleation is suppressed, conditions favorable to single-crystal growth are fostered

1.8 CONCLUSION

At this point we conclude this introductory sweep through several relevant topics in materials science If the treatment of structure, bonding, thermody- namics, and kinetics has introduced the reader to or elevated his or her prior awareness of these topics, it has served the intended purpose Threads 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 Both Au, which is FCC, and W, which is body-centered cubic (BCC) have a density of 19.3 g/cm3 Their respective atomic weights are 197.0

and 183.9

a What is the lattice parameter of each metal?

b Assuming both contain hard sphere atoms, what is the ratio of their diameters?

3 a Comment on the thermodynamic stability of a thin-film superlattice composite consisting of alternating Si and Ge,,,Si,., film layers shown

in Fig 14-17 given the Ge-Si phase diagram (Fig 1-12)

b Speculate on whether the composite is a single phase (because it is a single crystal) or consists of two phases (because there are visible film interfaces)

4 Diffraction of 1.5406-i X-rays from a crystallographically oriented (epitaxial) relaxed bilayer consisting of AlAs and GaAs yields two closely spaced overlapping peaks The peaks are due to the (1 11) reflections from

both films The lattice parameters are a,(AlAs) = 5.6611 A and a,(GaAs) = 5.6537 A What is the peak separation in degrees?

5 The potential energy of interaction between atoms in an ionic solid as a function of separation distance is given by V ( r ) = - A / r + Br-",

where A , B, and n are constants

a Derive a relation between the equilibrium lattice distance a, and A ,

B, and n

b The force constant between atoms is given by K , = d 2 V / d r 2 I r = l l o

If Young's elastic modulus (in units of force/area) is essentially given

by K , / a , , show that it varies as aG4 in ionic solids

6 What is the connection between the representations of electron energy in Figs 1-8a and 1-9? Illustrate for the case of an insulator If the material

in Fig 1-8a were compressed, how would E, change? Would the electrical conductivity change? How?

7 A 75 at% Ga-25 at% As melt is cooled from 1200 "C to 0 " C in a crucible

a Perform a complete phase analysis of the crucible contents at 1200 "C,

lo00 O C , 600 OC, 200 OC, 30 O C , and 29 'C What phases are present? What are their chemical compositions, and what are the relative amounts of these phases? Assume equilibrium cooling

Exercises 45

b A thermocouple immersed in the melt records the temperature as the crucible cools Sketch the expected temperature-time cooling re- sponse

c Do a complete phase analysis for a 75 at% As-25 at% Ga melt at lo00

' C , 800 "C, and 600 " C

8 A quartz (SiO,) crucible is used to contain Mg during thermal evapora- tion in an effort to deposit Mg thin films Is this a wise choice of crucible material? Why?

9 A solar cell is fabricated by diffusing phosphorous ( N dopant) from a constant surface source of lozo atoms/cm3 into a P-type Si wafer

containing 10l6 B atoms/cm3 The difisivity of phosphorous is

cm2/sec, and the diffusion time is 1 hour How far from the surface is the junction depth-i.e., where C , = C,?

10 A brass thin film of thickness d contains 30 wt% Zn in solid solution within Cu Since Zn is a volatile species, it readily evaporates from the

free surface ( x = d ) at elevated temperature but is blocked at the

a What is the vacancy formation energy?

b What fraction of sites will be vacant at 1080 "C?

12 During the formation of SiO, for optical fiber fabrication, soot particles

500 in size nucleate homogeneously in the vapor phase at 1200 "C If the surface energy of SiO, is loo0 ergs/cm2, estimate the value of the supersaturation present

13 An ancient recipe for gilding bronze statuary alloyed with small amounts

of gold calls for the following surface modification steps

(1) Dissolve surface layers of the statue by applying weak acids (e.g., vinegar)

46 A Review of Materiais Science

(2) After washing and drying, heat the surface to as high a temperature as possible but not to the point where the statue deforms or is damaged

(3) Repeat step 1

(4) Repeat step 2

( 5 ) Repeat this cycle until the surface attains the desired golden appear- Explain the chemical and physical basis underlying this method of gilding

M F Ashby and D R H Jones, Engineering Materials, Vols 1 and 2,

Pergamon Press, Oxford (1980 and 1986)

C R Barrett, W D Nix, and A S Tetelman, The Principles of Engineering Materials, Prentice Hall, Englewood Cliffs, NJ (1973)

0 H Wyatt and D Dew Hughes, Metals, Ceramics and Polymers,

Cambridge University Press, London (1974)

J Wulff, et al., The Structure and Properties of Materials, Vols 1-4,

Wiley, New York (1964)

M Ohring, Engineering Materials Science, Academic Press, San Diego

(1995)

L H Van Vlack, Elements of Materials Science and Engineering,

Addison-Wesley, Reading, MA (1989)

B Structure

2 C S Barrett and T B Massalski, The Structure of Metals, McGraw-Hill,

3 G Thomas and M J Goringe, Transmission Electron Microscopy of

References 47

2 A H Cottrell, Mechanical Properties of Matter, Wiley, New York

3 D Hull, Introduction to Dislocations, Pergamon Press, New York

(1964)

(1965)

D Classes of Solids

a Metals

1 A H Cottrell, Theoretical Structural Metallurgy, St Martin’s Press,

2 A H Cottrell, An Introduction to Metallurgy, St Martin’s Press, New

New York (1957)

York (1967)

b Ceramics

1 W D Kingery, H K Bowen, and D R Uhlmann, Introduction to

Ceramics, Wiley, New York (1976)

c Glass

1 R H Doremus, Glass Science, Wiley, New York (1973)

d Semiconductors

1 S M Sze, Semiconductor Devices-Physics and Technology, Wiley,

2 A S Grove, Physics and Technology of Semiconductor Devices,

3 J M Mayer and S S Lau, Electronic Materials Science: For Integrated

New York (1985)

Wiley, New York (1967)

Circuits in Si and GaAs, Macmillan, New York (1990)

E Thermodynamics of Materials

1 R A Swalin, Thermodynamics of Solids, Wiley, New York (1962)

2 C H Lupis, Chemical Thermodynamics of Materials, North-Holland,

New York (1983)

48 A Review of Materials Science

F Diffusion, Nucleation, Phase Transformations

1 P G Shewmon, Diffusion in Solids, McGraw-Hill, New York (1963)

2 J Verhoeven, Fundamentals of Physical Metallurgy, Wiley, New York

3 D A Porter and K E Easterling, Phase Transformations in Metals and

(1975)

Alloys Van Nostrand Reinhold, Berkshire, England (1981)

G Mathematics of Diffusion

1 H S Carslaw and J C Jaeger, Conduction of Heat in Solids, Oxford

2 J Crank, The Mathematics of Diffusion, Oxford University Press, University Press, London ( 1959)

London ( 1964)

Chapter 2

1

Vacuum Science and Technology

Virtually all thin-film deposition and processing methods as well as techniques

employed to characterize and measure the properties of films require a vacuum

or some sort of reduced-pressure environment For this reason the relevant

aspects of vacuum science and technology are discussed at this point It is also

appropriate in a broader sense because this subject matter is among the most

undeservedly neglected in the training of scientists and engineers This is

surprising in view of the broad interdisciplinary implications of the subject and

the ubiquitous use of vacuum in all areas of scientific research and technologi-

cal 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 KINETIC THEORY OF GASES

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 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 or pressure of the gas In the perfect or ideal gas approximation, there are no attractive or repulsive forces between molecules Rather, they may be considered to behave like independent elastic spheres separated from each other by distances that are large compared with their size The net result of the continual elastic collisions and exchange of kinetic energy is that a steady-state distribution of molecular velocities emerges given by the celebrated Maxwell-Boltzmann formula

(2-1)

2 R T '

This centerpiece of the kinetic theory of gases states that the fractional number

of molecules f ( v ) , where n is the number per unit volume in the velocity range v to u + dv, is related to their molecular weight (M) and absolute temperature (T) In this formula the units of the gas constant R are on a per-mole basis

Among the important implications of Eq 2-1, which is shown plotted in Fig 2-1, is that molecules can have neither zero nor infinite velocity Rather, the most probable molecular velocity in the distribution is realized at the maximum

v(1 o 5cm .s' )

Figure 2-1

sion from Ref 1)

Velocity distributions for A1 vapor and H, gas (Reprinted with permis-

2.1 Kinetic Theory of Gases 51

value of f ( u ) and can be calculated from the condition that df(u)/du = 0 Since the net velocity is always the resultant of three rectilinear components

u, , u,, , and u, , one or even two, but of course not all three, of these may be zero simultaneously Therefore, a similar distribution function of molecular velocities in each of the component directions can be defined; i.e.,

n dux I 2 r R T I 2 R T ’

f ( u x ) = = -

and similarly for the y and z components

A number of important results emerge as a consequence of the foregoing

equations For example, the most probable (urn), average (V), and mean square (u2) velocities are given, respectively, by

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 P is related to the mean-square velocity of the molecules and,

52 Vacuum Sclence and Technology

thus, alternatively to their kinetic energy or temperature Thus,

p = - - u 2 = -

NA NA ' where NA is Avogadro's number From the definition of n it is apparent that

Eq 2-4 is also an expression for the perfect gas law Pressure is the most widely quoted system variable in vacuum technology, and this fact has generated a large number of units that have been used to define it under various circumstances Basically, two broad types of pressure units have arisen

in practice In what we shall call the scientific system (or coherent unit system (Ref 2)), pressure is defined as the rate of change of the normal component of

momentum of impinging molecules per unit area of surface Thus, the pressure

is normally defined as a force per unit area, and examples of these units are dynes/cm2 (CGS) or newtons/meter2 (N/m2) (MKS) Vacuum levels are now commonly reported in SI units or pascals; 1 pascal (Pa) = 1 N/m2 Histori- cally, however, pressure was, and still is, measured by the height of a column

of liquid, e.g., Hg or H 2 0 This has led to a set of what we shall call practical

or noncoherent units such as millimeters and microns of Hg, torr, atmo- spheres, etc., which are still widely employed by practitioners as well as by equipment manufacturers Definitions of some units together with important conversions include

1 atm = 1.013 x IO6 dynes/cm2 = 1.013 x l o 5 N/m2 = 1.013 x l o 5 Pa

1 torr = 1 mm Hg = 1.333 x l o 3 dynes/cm2 = 133.3 N/m2 = 133.3 Pa

1 bar = 0.987 atm = 750 torr

The mean distance traveled by molecules between successive collisions, called the mean-free path &@, is an important property of the gas that depends on the pressure To calculate A,,,@,, we note that each molecule presents a target area ad: to others, where d , is its collision diameter A binary collision occurs each time the center of one molecule approaches within

a distance d , of the other If we imagine the diameter of one molecule increased to 2d, while the other molecules are reduced to points, then in traveling a distance A,,,@ the former sweeps out a cylindrical volume ?rdfA,,,,,

One collision will occur under the conditions rd f$ *n = 1 For air at 5oom temperature and atmospheric pressure, &@ = 500 A, assuming d , = 5 A

A molecule collides in a time given by A,,,@ / u and under the previous

conditions, air molecules make about 10'' collisions per second This is why

gases mix together rather slowly even though the individual molecules are