7.2.1 INTRODUCTION

Since they envelop and confine all solids including thin films and substrates, surfaces provide a key to understanding atomic behavior immediately beneath them as well as just above them. For many engineering purposes, thesefew atom layers collectively and seamlessly merge into what we call the “surface.” Real surfaces are usually contaminated with adsorbed gases and assorted compound layers. However, in this chapter surfaces will generally represent an atomically sharp interfacial demarcation between condensed-phase and gas-phase atoms.

There are many important reasons for studying the nature of clean surfaces. First of all, substrate surfaces usually provide the template for subsequent deposition and growth of thin films. In addition, surfaces also serve as the vehicle for imparting interfacial structural, mechanical, and physical properties to films. Secondly, assorted surface-analytical techniques are widely employed to reveal the chemical composition and structure of films. Through stimulation by photons, electrons, and ion beams, the electronic and physical structures of surface and near-surface atoms are altered in ways that can be detected, thus serving to fingerprint their identity and characterize their behavior. Thin films also confront the environment and interact chemically, mechanically, electrically, etc., with the ambient through their surfaces. And, as film thicknesses continue to shrink, which is the trend in many applications, a larger fraction of atoms will reside at the top film surface and bottom film–substrate interface; these surface atoms will then play increasingly greater roles in influencing the behavior of the atoms in between.

In this section we shall start by treating both the electronic and physical structures of subsurface atom layers of solids, and in particular, substrates. Then we will consider geometric placement of different atoms in the surface plane itself, our main interest being the first atomic layer of the film that deposits above the substrate surface plane. All in all perhaps only 15 Å of material at most is involved. Amazing as it seems, a profile through this vanishingly small thickness reveals that a minimum of two chemically different species interact in a continuum of bonding arrangements, and that at least three crystallographic structures can be differentiated. It is a testament to the great advances in surface science experimentation and theory that have enabled such distinctions to be made. An excellent review of the progress in this area to date is detailed in Surface Science: The First Thirty Years (Ref. 3). Importantly, through the electronic and optoelectronic devices it has made possible, surface science is intimately connected to the story of a “… technological revolution which is changing the world as we know it, transforming a planet into a global village.”

7.2.2 ELECTRONIC NATURE OF SURFACES

7.2.2.1 Metals

Solid-state physics has long considered the quantum nature of bulk crystalline solids. By now we are all familiar with the notion that the periodic electrostatic potential in lattices is responsible for electron energy bands and gaps, as well as the classification of solids into metals, semiconductors, and insulators. Although solutions to the Schrodinger wave equation for the many-body periodic potential are rarely possible, useful approximations exist. For example, assorted theoretical models, e.g., free electron, nearly free electron, tight binding, and pseudopotential, have yielded wave functions and energies for electron states in metals (Ref. 4). The electronic structure is commonly depicted by the band diagram shown in Fig. 1-9a, a representation applicable to bulk regions of the metal. At the surface, however, the assumed uniform positive potential due to ion cores is abruptly terminated. As a consequence, the mobile electrons within the bulk attempt to screen out the now unbalanced attractive potential at the surface. This results in an increased electron density at the surface which dies away rapidly in an oscillatory manner within the interior (Ref. 5). In solids with low electron densities the charge oscillations die away more slowly. Some of these features are shown in Fig. 7-3, where electron charge–density contours are plotted for nickel. At the surface the charge density is smeared out but it becomes periodic as the interior ion cores are reached. This smoothing of the surface charge is the compromise electrons strike when they lower both their potential and kinetic energies by respectively penetrating into the interior and by spreading out toward the surface.

Figure 7-3 Electron charge density contours around Ni (001) atoms near the surface.

(After F. J. Arlinghaus, J. G. Gay, and J. R. Smith, Phys. Rev. B21, 2055, (1980). Reprinted with the permission of John R. Smith.)

Surface or Tamm states were first predicted based on Schrodinger equation solutions to electrons in terminated periodic lattices. From the standpoint of quantum mechanics, the loss of three-dimensional lattice periodicity at the surface allows for quantum states that lie within the (bulk) energy gap (Ref. 6). Localized surface states are characterized by waves that travel parallel to the surface but not into the solid. When electron or hole charge occupies these states, electrostatic fields penetrate the interior and produce a variation of potential that is manifested as band bending. Adsorbed impurities, emergent dislocations, and surface relaxation (or reconstruction; see Section 7.2.4) also create surface states. But in all cases the extent of band bending in metals is generally negligible because there are sufficient numbers of mobile electrons to screen out or compensate for the surface-state charge; this is generally not true in semiconductors.

The work function (φ), an important surface property, is a consequence of the foregoing model. We may view φ as related to the difference in electron potential between that associated with the surface-charge relative to the average value within the bulk lattice. Thus q φ where q is the electronic charge, is the energy barrier electrons must surmount when leaving the solid. Measured values for φ in most elements cluster around 3.5 ± 1.5 V. Contributing to φis a charged double layer generated by the electron asymmetry at the surface. Alteration of this double layer due to adsorbed contaminants, alloying, or the crystallographic face exposed typically changes the magnitude of φby a few tenths of a volt.

7.2.2.2 Semiconductors

There are significant structural as well as electronic differences between semiconductor and metal surfaces. This fact is immediately apparent if we consider the interface between a (111), III–V compound semiconductor surface and vacuum shown in Fig. 7-4a. Forgetting distinctions between III–and V-type atoms for the moment, this ball-and-stick model is applicable to elemental semiconductors such as (111) silicon as well. The severed covalent bonds at the surface mean the presence of uncompensated charge and an electric field. However, dangling bonds from neighboring surface atoms bridge to complete the bonding requirements. In the process surface atoms shift position and the electrostatic energy in the vacuum is reduced. As we shall soon see (Sections 7.2.3.1 and 7.2.4.1), the surface reconstructs with important structural and property implications.

Figure 7-4a Model of (111) surfaces in the zinc blende crystal structure of a III–V compound semiconductor.

(From H. C. Gatos, Surf. Sci. 299/230, 1, (1994). Reprinted with the permission of Professor Harry C. Gatos, MIT.)

The same thing happens at III–V semiconductor surfaces but with interesting added twists. As the figure indicates, the surface consists entirely of Group V atoms (or light B atoms, e.g., As), but it could just as easily consist of Group III atoms (or dark A atoms, e.g., Ga). These unique and distinct surfaces arise because cuts can only be made between planes AA and BB. The reasons for such cuts are: (a) only one bond between atoms needs to be disrupted and (b) the surface atoms remain triply bonded to the lattice. Other cuts such as those between AA and B’B’ are not possible because three bonds must be severed and resultant surface atoms are singly bonded to the lattice. For the BB, so-called (111) surface, we note that only three of five valence electrons in each arsenic atom are attached or bonded, while on an AA (111) surface all three of boron’s valence electrons are bonded. As a consequence, the As surface is more electronically active than the Ga surface, and can, for example, be smoothed and polished more readily by etching. In contrast, Ga surfaces etch slowly and tend to be rougher. Such differences lead to varied epitaxial phenomena when films are deposited on these surfaces.

Unlike metals, two bands–a lower valence as well as an upper conduction band–play important roles in semiconductors and devices based on them. Another difference is the band bending that occurs at semiconductor surfaces that contact other materials such as insulators, as shown in Fig. 7-4b. Band bending or the absence of a flat band condition means that a nonzero electrical potential ψ_s exists at the surface. What generates ψ_s are the severed covalent bonds at the semiconductor surface which are essentially localized surface-states that trap electrons or holes. Together with ionized donors or acceptors they create a spatially varying charge distribution near the surface. The situation parallels that of the plasma-sheath region (Section 4.3.5). A solution to the Poisson–Boltzmann equation in one dimension essentially yields a form of ψ similar to that for the electrostatic potential in a plasma (e.g., Eq. 4-17, also problem 5 in Chapter 4). Because there are relatively few carriers to shield surface charge, band bending is much more pronounced in semiconductors than in metals, e.g., the electrostatic potential typically penetrates ∼ 1 μm below the surface. A space charge region is said to exist where the bands are bent. Within it the semiconductor exhibits dielectric behavior and supports the internal electric-field, i.e., that makes possible rectifying action at a p–n junction.

Figure 7-4b Energy band diagram at the surface of a p-type semiconductor. The potential ψ is defined to be zero in the bulk but assumes a nonzero value at the surface that depends on the phase contacting the semiconductor, e.g., vacuum, SiO₂, or other semiconductor.

(After S. M. Sze, Physics of Semiconductor Devices, 2nd ed. John Wiley & Sons, New York, 1981.)

For semiconductor surfaces to perform electrical functions in devices they are often contacted by either insulating, dielectric, or conducting thin films.

In silicon, for example, these films are invariably either grown or deposited SiO₂, and deposited metals (M) that serve as electrical contacts. Because Si–SiO₂ and M–Si structures are of such great technical importance, these interfaces are perhaps the most studied in all of physical science. As a result there is a broad accessible literature on interfacial Si–SiO₂ charge and barriers to electron flow across M–Si contacts; for this reason these topics will not be discussed here.

7.2.3 SURFACE STRUCTURES

7.2.3.1 Introduction

As we have already seen, minimization of the overall electronic and bonding energies in the topmost atom layers causes the surface crystallography to differ from that of the bulklike interior as suggested by the schematic cross-sectional views of Fig. 7-5. If the surface structure is the predictable extension of the underlying lattice, we have the case shown in Fig. 7-5a. However, the loss of periodicity in the vertical direction alters surface electronic properties and leaves energetic broken bonds. It is then more likely that the structure shown in Fig. 7-5b will prevail. The absence of bonding forces to underlying atoms results in new equilibrium positions which deviate from those in the bulk lattice. A disturbed surface layer known as the “selvedge” may then be imagined. Within this layer the atoms relax in such a way as to preserve the symmetry of the bulk lattice parallel to the surface, but not normal to it. A more extreme structural disturbance is depicted in Fig. 7-5c, where surface atoms rearrange into a structure with a symmetry that differs from that of the bulk solid. This phenomenon is known as reconstruction and can alter many surface structure-sensitive properties, e.g., chemical, electrical, optical, and sorption behavior.

Figure 7-5 Schematic cross-sectional views of close-packed atomic positions at a solid surface. (a) Bulk exposed plane. (b) Atomic relaxation outward. (c) Reconstruction of outer layers.

(From Ref. 5, © Oxford University Press, by permission.)

Whereas atomic positioning in the selvedge cross-section is not amenable to simple analysis, the crystallography in the surface plane is, and we now proceed to discuss it. In particular, we are interested in such surfaces covered by a periodic array of adsorbed atoms (adatoms).

7.2.3.2 Crystallography of Substrate Meshes and Adatom Arrays

Just as there are 14 Bravais lattices in three dimensions, so there are just five unit meshes or nets, corresponding to a two-dimensional surface as shown in Fig. 7-6. Points representing atoms may be arranged to outline (1) squares, (2) rectangles, (3) centered rectangles, (4) rhombi, and (5) parallelograms with included angles of 60° and 120°. Miller-type indices are used to denote atom coordinates, directions, and distances between lines within the surface.

Figure 7-6 The five diperiodic surface nets.

Consider a mesh of substrate atoms that is covered with adatoms arrayed in a monolayer pattern. The methodology for identifying the structural geometry employs simple vector concepts (Refs. 5, 7). Vectors describing the adsorbate positions (b) are simply related to those of the substrate lattice (a). Possessing components b_i and a_i, with i = 1,2 denoting the x and y directions, we may generally write

     (7-1a)

     (7-1b)

Terms M_ij denote the components of the transformation matrix between adatom and substrate coordinates. Let us apply these equations to Fig. 7-7a where a square monolayer pattern of adatoms denoted by () covers a net of substrate atoms represented by • (dots). This combination might correspond to the early growth of an epitaxial layer on the (100) plane of a BCC or FCC crystalline substrate; for the latter, however, a larger face-centered cell must be chosen. In the case shown, b₁ = 2a₁ and b₂ = 2a₂.

Figure 7-7 Atomic positions of adatoms () relative to substrate atoms • arrayed on a (100) simple cubic plane. (a)P(2 × 2) overlayer structure. (b)C(2 × 2) centered lattice overlayer or structure. (c)P(1 × 2) overlayer structure. (d) P(2 × 1) overlayer structure. Corresponding reciprocal lattices for (c) and (d) are shown (x refers to adatoms).

Thus

and we speak of a P(2 × 2) overlayer structure, where P indicates a primitive unit cell effectively containing only one atom. If, for example, this structure physically corresponds to a deposit of sulfur atoms on (100) Fe, the notation would be Fe(100) P(2 × 2)-S. In a similar vein, suppose there is an additional adatom positioned centrally as shown in Fig. 7-7b. Intuition would suggest that such a structure be identified asC(2 × 2) where C denotes the centered lattice. The matrix notation for the dashed primitive cell in this case is

Because the overlayer primitive cell dimensions are times larger and rotated relative to substrate unit cell, an alternative notation is where R stands for rotation.

Consider now the repetitive rectangular adatom cell pattern in Fig. 7-7c where b₁ =a₁ and b₂ = 2a₂. Therefore,

and the overlayer is denoted by P(1 × 2). Similarly, Fig. 7-7d shows an overlayer of the same geometry but rotated 90° with respect to case c; the notation would be P(2 × 1).

It should be noted that there is a certain ambiguity in identifying overlayers. We have only considered caseswhere the adatoms fall directly above substrate atoms. But adatoms may also lie in pockets between substrate-atom sites. This can happen, for example, by shifting each adatom by ½a₁ and ½a₂. But vector operations are independent of the choice of origin and such overlayers would be identified by exactly the same notations as before, e.g., P(2 × 2),P(1 × 2).

7.2.4 RECONSTRUCTED SILICON SURFACES

7.2.4.1 Crystallography of (001) Si

On a clean (001) Si surface, atoms are surprisingly not arranged according to the much reproduced bulk diamond-cubic lattice structure where the nearest-neighbor atom separation is 0.385 nm. Instead, the isotropic face-centered square array of Si atoms assumes an anisotropic reconstructed structure of lower symmetry (Ref. 8) possessing two characteristic directions. These lie in the surface plane either along or perpendicular to dimer (paired atom) rows in orthogonal 〈110〉 orientations. The parallel dimer rows aligned at 135° with respect to the page horizontal, are easily visible in the STM image of Fig. 7-8a.

Figure 7-8 (a) Scanning tunneling microscope image of parallel dimer rows.

(From Ref. 35.) (b) Schematic side view of the reconstructed (001) Si surface having a (2 × 1) structure. (c) Top view of reconstructed (2 × 1) Si surface. The intensified unit cell measures 0.385 nm × 0.770 nm. (From Ref. 8.) (Reprinted with the permission of Max G. Lagally.)

More will be said about STM methods in Chapter 10. Suffice it to say that in this technique a metal probe tip sharpened to atomic dimensions is brought to within a few angstroms of the surface and scanned over its periodically bumpy atomic terrain, much like a stylus over a record. The current that flows between probe and surface reflects the topography of individual atoms on surfaces as well as the geometry of terraces and ledges or steps on the substrate surface.

Schematically shown in both side and top views of Figs. 7-8b and 7-8c is the (2 × 1) reconstructed surface structure on (001) Si. In these figures the position of silicon atoms, differentiated by circle size, is shown at three successive levels. Within dimers the Si atoms would like to merge even closer together but are restrained from doing so by surrounding atoms. Therefore, a tensile stress develops in the dimer-bond direction; similarly along the dimer row a smaller tensile or perhaps compressive stress is expected. Because of stress anisotropy the reconstructed surface of (001) Si decomposes into two alternating (2 × 1) terrace-like domains with monatomic steps at the boundaries. Dimer rows rotate 90° as successive terraces are accessed giving rise to two types of monatomic steps. The so-called S_A step is parallel to upper-terrace dimer rows, while S_B steps are perpendicular to these same rows. Because it is difficult to disrupt the linearity of dimer rows, the energy to form a kink in an S_A step is higher than for an S_B step. Thus S_A steps tend to appear smooth while S_B steps are rough. What happens when Si atoms deposit on such a terraced substrate is discussed in Section 7.5.4.1.

7.2.4.2 Crystallography of (111) Si

Reconstruction of (111) Si surfaces into (7 × 7) structures was first observed in LEED patterns (Ref. 9) similar to those shown in Fig. 8-33a. Disbelieved at first, these beautiful patterns were attributed to contaminated surfaces. It was some 25 years later that the now ubiquitous STM image of (111) Si shown in Fig. 7-9 independently confirmed the existence of the (7 × 7) reconstruction. Unfolding the surface and subsurface atomic positions that give rise to such a complex crystallographic structure is a formidable task. A minimum of 49 atom positions are involved and since the reconstruction occurs over several layers many more atom coordinates (perhaps hundreds) must be specified. Detailed crystallographic models for (7 × 7) Si have been proposed. This subject is clearly beyond our scope and therefore, the interested reader is referred to Refs. 10 and 11.

Figure 7-9 Figure 7-9 Scanning tunneling microscope image of the reconstructed (7 × 7) surface of (111) Si.

(Courtesy of Y. Kuk, AT&T Bell Laboratories.)

We shall return to the reconstructed (001) and (111) Si surfaces at the end of the chapter where the early stages of film nucleation on them areconsidered.

7.2.5 Adsorption Reactions on Solid Surfaces

Let us now take the pristine substrate surfaces we have been considering and expose them to gas particles. This is what occurs, for example, in thin-film deposition processes or during the operation of catalysts (Ref. 12). The first thing that happens is surface adsorption–the process in which impinging atoms and molecules enter and interact within the transition region between the gas phase and substrate surface (Ref. 13). Two kinds of adsorption processes, namely physical (physisorption) and chemical (chemisorption), can be distinguished depending on the strength of the atomic interactions. If the particle (molecule) is stretched or bent but retains its identity, and van der Waals forces bond it to the surface, then we speak of physisorption. However, chemisorption is involved whe particle changes its identity through ionic or covalent bonding with substrate atoms. The two sorptions can be quantitatively distinguished on the basis of adsorption heats or energies, E_p and E_c, for physisorption and chemisorption, respectively. Typically E_p ∼0.25 eV while E_c ∼1–10 eV.

Individual physi- and chemisorption processes can be visualized in the respective curves P and C of Fig. 7-10, representing the potential energies of interaction as a function of distance r(see Fig. 1-8). It is evident that physisorbed particles achieve equilibrium further from the surface (i.e., at r = r_p) than chemisorbed species (at r = r_c). In addition to E_p and E_c we must consider the dissociation energy of molecules, E_dis. If E_dis is less than E_c, dissociative chemisorption is thermodynamically possible. When both adsorption processes occur the curves merge and assume a roller-coaster shape with an effective energy barrier E_a that governs the rate of the transition from the physisorbed to chemisorbed states. In the subsequent treatment of film nucleation a composite energy of desorption (E_des) that integrates over all desorption states will be used. Similarly, an energy of adsorption (E_ads) related to either E_p or E_c, or some combination of these, can be defined.

Figure 7-10 Model of physisorption and chemisorption processes in terms of the interaction potential between adsorbate and surface vs distance.

7.3.1 SCOPE

Paradoxically, nucleation events which may only involve very few atoms can be profitably described in terms of classical thermodynamics, a subject that deals with the continuum behavior of many atoms. Among the questions addressed in this section are:

However, issues related to atomic-scale nucleation events, the rates at which they occur, and the numbers of stable nuclei produced per unit time are not easily amenable to thermodynamics and will be treated in subsequent sections. Since surface energies of materials play a key role in these discussions, this subject is a good place to start.

7.3.2 SURFACE ENERGIES

In the well-known capillarity or droplet theory of homogeneous nucleation, a solid nucleates from a prior unstable liquid or vapor phase by respectively establishing a solid–liquid (s-l) or solid–vapor (s-v) interface. Associated with these interfaces or surfaces are corresponding interfacial energies. Atoms at free surfaces are more energetic than atoms within the underlying bulk because they make fewer bonds with surrounding atoms and are thus less constrained. The difference in interatomic energy of atoms at these two locations is the origin of surface energy. Alternatively, there is a thermodynamic driving force to reduce the number of necessarily cut, dangling bonds at the surface through rebonding between atoms. We may then equivalently view the surface energy γ (J/m²) in terms of the energy reduction per unit area.

In a recommended discussion of surface energies of solids by Tu et al. (Ref. 17) a number of issues relevant to surfaces are raised. With respect to material classes and systematics, metals have higher surface energies than oxides, alkali halides, sulfides, and organic substances. For virtually all elements in the liquid and solid states the surface energies roughly span the range 0.2 to 3 J/m² with 1 J/m² (= 1000 erg/cm²) being typical. The higher values are associated with the transition metals while the lower values are indicative of alkali and divalent metals, metalloids, and inert gases. For organic and aqueous solutions γ is often less than 100 mJ/m². Furthermore, the surface energy drops slightly with temperature T; typically, dγ/dT ∼ −0.05 mJ/m²-°C.

Surface energies are often interchangeably called surface tensions. These terms are not strictly identical although units of the latter, i.e., force/length, are equivalent to energy/area. The basic definition of surface energy is related to the reversible work (dW) on a material when its surface area (A) is increased, or dW = γdA. Surfaces can alter their energies through change in either A or γ. The latter possibility implies altering atomic arrangements or reconstructing the surface and may be thought of as the work in mechanically stretching or straining the surface. For solids, surface stresses are thus involved, but these vanish in liquids because they cannot support strains. It is the shortening of the surface bond length plus the effective conversion of bulk to surface bonds that essentially causes liquid mercury to ball up; in the process A is reduced. We shall make no further distinction between surface energy and surface tension.

7.3.3 CAPILLARITY THEORY OF HETEROGENEOUS NUCLEATION

We have already considered a simple model for homogeneous nucleation of a condensed phase from a prior supersaturated vapor (see Section 1.7). Homogeneous nucleation, although rare in other contexts, occurs with some frequency in CVD reactors where too highly supersaturated gases rain “snow” down on substrates. The process is easy to model since it occurs without benefit of the complex heterogeneous sites that exist on an accommodating substrate surface. In a similar spirit we shall extend this capillarity theory to the case of heterogeneous nucleation of a condensed film on a substrate. Capillarity theory (Refs. 17-19) possesses the mixed virtue of yielding a conceptually simple qualitative model of film nucleation which is, however, quantitatively inaccurate. The lack of detailed atomistic assumptions gives the theory an attractive broad generality with the power of creating useful connections among such variables as substrate temperature, deposition rate, and critical film nucleus size.

We start by assuming that film forming atoms or molecules in the vapor phase impinge on the substrate creating nuclei of mean dimension r. The free-energy change accompanying the formation of such an aggregate is given by

     (7-6)

As in homogeneous nucleation the chemical free-energy change per unit volume, ΔG_v, drives the condensation reaction. There are several interfacial tensions, γ, to contend with now and these are identified by the subscripts f, s, and v representing film, substrate, and vapor, respectively. A pair of subscripts refers to the interface between the indicated phases. For the spherical cap-shaped solid nucleus shown in Fig. 7-11 the curved surface area (a₁r²), the projected circular area on the substrate (a₂r²), and the volume (a₃r³) are involved, and the respective geometric constants are a₁ = 2π(1 − cos θ), a₂ = π sin²θ, and .

Figure 7-11 Schematic of basic atomistic nucleation processes on substrate surface during vapor deposition.

The mechanical equilibrium among the horizontal components of the interfacial tensions or forces surrounding the nucleus yields Young’s equation,

     (7-7)

Importantly, the contact or wetting angle θ, which figures prominently in the formulas for r^* and ΔG^* developed later, depends solely on the surface properties of the involved materials.

Returning now to Eq. 7-6, we note that any time a new interface appears there is an increase in surface free energy and hence the positive sign for the first two surface terms. Similarly the loss of the circular substrate–vapor interface under the cap implies a reduction in system energy and a negative contribution to ΔG. Thermodynamic equilibrium is achieved when dΔG/dr = 0, which occurs at a critical nucleus size r = r^* given by

     (7-8)

Correspondingly, ΔG evaluated at r = r^* is

     (7-9)

Both r^* and ΔG^* scale in the manner shown in Fig. 1-20. An aggregate smaller in size than r^* disappears by shrinking, lowering ΔG in the process. Critical nuclei grow to supercritical dimensions by further addition of atoms, a process which lowers ΔG still more. In heterogeneous nucleation the accommodating substrate catalyzes vapor condensation by lowering the energy barrier ΔG^* through a reduction of the contact angle. After substitution of the geometric constants, it is easily shown that ΔG^* is essentially a product of two factors, i.e.,

     (7-10)

The first is the value for ΔG^* derived for homogeneous nucleation. It is modified by the bracketed term, a wetting factor dependent on θ which has the value of zero for θ = 0 and unity for θ = 180°. When the film wets the substrate, there is no barrier to nucleation. At the other extreme of dewetting, ΔG^* is maximum and equal to that for homogeneous nucleation.

The preceding formalism provides a generalized framework for inclusion of other energy contributions. If, for example, the film nucleus is elastically strained throughout because of the bonding mismatch between film and substrate, then a term a₃r³ΔG_s, where ΔG_S is the strain free-energy change per unit volume, would be appropriate. In the calculation for ΔG^*, the denominator of Eq. 7-9 would then be altered to (ΔG_V+ ΔG_S)2. Because the sign of ΔG_V is negative while ΔG_S is positive, the overall energy barrier to nucleation increases in such a case. If, however, deposition occurred on an initially strained substrate, i.e., one with emergent cleavage steps or screw dislocations (see Fig. 8-10), then stress relieval during nucleation would be manifested by a reduction of ΔG^*. Substrate charge and impurities would similarly influence ΔG^* by affecting terms related to either surface and volume electrostatic, chemical, etc., energies.

The most important thing to remember about ΔG^* is its strong influence on the density (N^*) of stable nuclei that can be expected to survive. Based on our previous experience (Eq. 1-3) of associating the probable concentration of an entity with its characteristic energy through a Boltzmann factor, it is appropriate to take

     (7-11)

where n_s is the total nucleation site density. We shall return to this equation subsequently.

7.3.4 FILM GROWTH MODES

Young’s equation (Eq. 7-7) provides a way to distinguish and better understand the three modes of film growth displayed in Fig. 7-1. For island growth, θ > 0, and therefore,

     (7-12)

If γ_fs is neglected, this relation suggests that island growth occurs when the surface tension of the film exceeds that of the substrate. This is why deposited metals tend to cluster or ball up on ceramic or semiconductor substrates.

In the case of layer growth the deposit “wets” the substrate and θ 0. Therefore,

     (7-13)

A special case of this condition is ideal homo- or “auto-epitaxy.” Because the interface between film and substrate essentially vanishes, γ_fs = 0. High-quality epitaxial films require the avoidance of any disruption to layer growth. Superlattices composed of alternating A and B epitaxial films ordered in a stack represent a particularx challenge in this regard. Sequential depositions mean A deposited on a B film substrate followed by B deposited on an A film substrate. This stacking asymmetry is often not a problem when A and B are heteroepitaxial compound semiconductors; high-performance lasers have been fabricated in this way. Although roughly equal surface energies are involved here, this is not usually the case in metal–metal or metal–semiconductor superlattices. In general, materials with low surface energy will wet substrates with a higher surface energy. Thus Benjamin Franklin observed that a little oil smoothed the surface roughness of a large water pond. In this case the very thin oil film that formed dramatically modified the surface properties of water.

Lastly, for S-K growth, initially at least,

     (7-14)

In this case the strain energy per unit area of the film overgrowth is large with respect to γ_fv permitting nuclei to form above the initial layers. The transition from two- to three-dimensional growth which typically occurs after 5–6 monolayers is not completely understood; any factor which disturbs the monotonic decrease in binding energy characteristic of layer growth may be the cause. For example, because of film–substrate lattice mismatch, strain energy accumulates in the growing film. When released, the high energy at the deposit intermediate-layer interface may trigger island formation.

7.3.4.1 Morphological Stability of Strained Layers

To address strain (free) energy (G_S) semiquantitatively we note that elasticity theory (Eq. 1-45) suggests G_S = ½Yε² where Y is the elastic modulus of a layer and ε is its strain. A useful measure of ε at the interface between a film and substrate is the lattice mismatch strain or simply misfit, , defined as , where a₀ is the lattice parameter and f and s refer to the film and substrate, respectively. As we shall see, the quantity assumes an important role in epitaxy (Chapter 8).

Rather than view the breakdown of epitaxy due to dislocation generation as in Section 8.3.1, let us consider the critical film thickness beyond which a planar film surface roughens due to the growth of islands. This creates a morphological instability that can be treated in terms of macroscopic heterogeneous nucleation theory. Following Wessels (Ref. 20), the net free energy change for the nucleation of a hemispherical island (not spherical cap) on top of a growing, strained epitaxial layer of thickness h is given by

     (7-15)

In the process the island is assumed to be incoherent (or relaxed) while the epilayer partially relaxes. By reference to Eq. 7-6, the volume free energy of the island and interfacial and strain energy interaction between the island and epitaxial layer are, respectively, ΔG_V, γ, and ΔG_S. The term ΔG_S is interpreted as the difference in epilayer strain energy per unit area after the island nucleates (½Yε²h) relative to that in the epilayer prior to island nucleation (). Therefore, where ε is the mean misfit in the strained film and A (= πr²) is the area affected by strain. As before, the critical nucleus radius (r^*) is determined by the condition that dΔG/dr = 0. Therefore,

     (7-16)

and in the limit that r^* = 0, the critical film thickness (h^*) for the onset of the rough island morphology (e.g., S-K growth) is

     (7-17)

This relation implies that h^* varies approximately as . For InGaAs on GaAs, Wessels suggests that holds after substituting appropriate constants. When there is a transition from 2-D layer to 3-D island film growth.

Now we may better understand Fig. 7-12, which distinguishes the regimes of influence for the three film growth modes in terms of surface energy ratio W(W = (γ_s − γ_f)/ γ_s) and . As noted earlier, when γ_f > γ_s, i.e., W < 0, island growth dominates. But its range of dominance expands when there is additional misfit present. Layer growth is possible only when W > 0; surprisingly, however, this growth mechanism can tolerate a small amount of misfit, a fact that makes strained-layer epitaxy feasible. In between island and layer film morphologies, and competing with them, is the S-K growth mode.

Figure 7-12 Stability regions of the three film growth modes in coordinates of surface energy difference between film and substrate (vertical) and lattice misfit (horizontal).

(From K. N. Tu, J. W. Mayer, and L. C. Feldman, Electronic Thin Film Science for Electrical Engineers and Materials Scientists, Macmillan, 1992. Originally published in IBM J. Res. Develop. Reprinted with the permission of Professor K. N. Tu.)

7.3.5 Nucleation Dependence on Substrate Temperature and Deposition Rate

Substrate temperature and deposition rate (atoms/cm²-s) are among the chief variables affecting deposition processes. Calculating their effect on r*and ΔG* is both instructive and simple to do. First we assume that is proportional to P_V in Eq. 1-38 so that

     (7-18)

where is the equilibrium evaporation rate from the film nucleus at the substrate temperature. Assuming an inert substrate, i.e., γ_fv = γ_fs, direct differentiation of Eq. 7-8 yields (Ref. 18)

     (7-19)

Assuming typical values, γ_fv = 1 J/m² and ∂γ_fv/∂T = −0.05 mJ/m²-K. An estimate for ∂ΔG_V/∂T is the entropy change for vaporization, which is roughly 8 × 10⁶ J/m³-K for many metals. As long as direct substitution shows that

     (7-20)

Similarly, by the same assumptions and arguments

     (7-21)

It is also simple to show that

     (7-22)

Direct chain-rule differentiation using Eqs. 7-8 and 7-18 yields

7-23)

Since ΔG_V is negative, the overall sign is negative. Similarly, it is easily shown that

     (7-24)

The preceding four inequalities have interesting implications and summarize a number of common effects observed during film deposition. From Eq. 7-20 we note that a higher substrate temperature leads to an increase in the size of the critical nucleus. Also, a discontinuous island structure is predicted to persist to a higher average coverage than at low substrate temperatures. The second inequality (Eq. 7-21) suggests that a nucleation barrier may exist at high substrate temperatures whereas at lower temperatures it is reduced in magnitude. Also, because of the exponential dependence of N* on ΔG*, the number of supercritical nuclei decreases rapidly with temperature. Thus, a continuous film will take longer to develop at higher substrate temperatures. From Eq. 7-22 it is clear that increasing the deposition rate results in smaller islands. Because ΔG* is also reduced, nuclei form at a higher rate suggesting that a continuous film is produced at lower average film thickness. If for the moment we associate a large r* and ΔG* with being conducive to large crystallites or even monocrystal formation, such conditions prevail at high substrate temperatures and low deposition rates. Alternately low substrate temperatures and high deposition rates yield polycrystalline deposits. These simple guidelines summarize much practical deposition experience in a nutshell.

A plot of these structural zones as a function of deposition rate and substrate temperature is shown in Fig. 7-13 for Cu films deposited on a (111) NaCl substrate. Maps of this sort have been graphed for other film–substrate combinations. In semiconductor systems a regime of amorphous film growth is additionally observed when is large and T is low.

Figure 7-13 Dependence of microstructure on deposition rate and substrate temperature for Cu films on (111) NaCl. Some substrate temperatures are noted.

(Reprinted with permission from the Metals Society. From Ref. 2, courtesy of R. W. Vook.)

Despite the qualitative utility of capillarity theory, it is far from being quantitatively correct. For example, let us estimate the value of r* for a typical metal film at 300 K. To simplify matters the homogeneous nucleation formula, r* = −2γ/ΔG_v, will be used where ΔG_v = −k_BT/Ωln P_v/P_s. Assuming Ω = 20 × 10⁻³⁰/m³, γ = 1 J/m², P_v = 10⁻³ torr, and P_s = 10⁻¹⁰ torr, then r* = −2(1.0)/−{[1.38 × 10⁻²³ × 300/20 × 10⁻³⁰]ln[10⁻³/10⁻¹⁰]} or 0.6 nm.

A nucleus of this size contains just a few atoms, and it is therefore doubtful that continuum concepts such as surface tension and nucleus radius have much significance. It is more realistic to interpret vapor condensation phenomena in terms of a heterogeneous nucleation theory based on atomistic modeling. This is one of the topics considered in the next section.

7.4.1 SCOPE

Among the questions addressed in this section are:

In particular, distinctions between the macroscopic (e.g., capillarity) and microscopic (atomistic) views of these processes will be stressed. For the most part theoretical modeling is emphasized.

7.4.2 NUCLEATION RATE

The nucleation rate is a convenient synthesis of terms that describes how many nuclei of critical size form on a substrate per unit area, per unit time. Nuclei can grow through direct impingement of gas-phase atoms, but this is unlikely in the earliest stages of film formation when nuclei are spaced far apart. Rather, the rate at which critical nuclei grow depends on the rate at which adsorbed monomers (adatoms) attach to it. In the model of Fig. 7-11 energetic vapor atoms which impinge on the substrate may immediately desorb but usually they remain on the surface for a length of time τ_s given by

     (7-25)

The vibrational frequency of an adatom on the surface is ν (typically 10¹³ s⁻¹) while E_des is the energy required to desorb it back into the vapor. Adatoms, which have not yet thermally accommodated to the substrate, execute random diffusive jumps and may, in the course of their migration, form pairs with other adatoms or attach to larger atomic clusters or nuclei. When this happens it is unlikely that these atoms will return to the vapor phase. This is particularly true at substrate heterogeneities such as cleavage steps or ledges where the binding energy of adatoms is greater relative to a planar surface. The proportionately large numbers of atom bonds available at these accommodating sites leads to higher E_des values. Thus a significantly higher density of nuclei is usually observed near cleavage steps and other substrate imperfections. The presence of impurities similarly alters E_des in a complex manner dependent on the type and distribution of atoms or molecules involved.

We now exploit some of these microscopic notions by noting that the nucleation rate is essentially proportional to the product of three terms, namely,

     (7-26)

N* is the equilibrium concentration (per cm²) of stable nuclei (Eq. 7-11) and ω is the rate at which atoms impinge (per cm²-s) onto the nuclei of critical area A* (cm²). Of the total nucleation site density n_s, a certain number will be occupied by adatoms. The surface density of these adatoms, n_a, is given by the product of the vapor impingement rate (Eq. 2-8) and adatom lifetime or

     (7-27)

Surrounding the cap-shaped nucleus of Fig. 7-11 are adatoms poised to attach to the circumferential belt whose area is

     (7-28)

Quantities r*, a₀, and θ were defined previously.

Lastly, the impingement rate onto area A* requires adatom diffusive jumps on the substrate with a frequency given by ν exp − (E_S/k_BT), where E_S is the activation energy for surface diffusion. The overall impingement flux is the product of the jump frequency and n_a or

     (7-29)

There is no dearth of adatoms which can diffuse to and be captured by the existing nuclei. During their residence time adatoms are capable of diffusing a mean distance X from the site of incidence given by

     (7-30)

The surface diffusion coefficient, D_s, is essentially (see Eq. 1-31)

     (7-31)

and therefore,

     (7-32)

Large values of E_des coupled with small values of E_s serve to extend the nucleus capture radius.

Upon substitution of Eqs. 7-11, 7-27, 7-28, and 7-29 in Eq. 7-26, we finally obtain

     (7-33)

The nucleation rate is a very strong function of the nucleation energetics which are largely contained within the term ΔG*. It is left to the reader to develop the steep dependence of on the vapor supersaturation ratio. As noted earlier, a high nucleation rate encourages a fine-grained or even amorphous structure while a coarse-grained deposit develops from a low value of .

7.4.3 ATOMISTIC MODELS OF THE NUCLEATION RATE

Atomistic theories of nucleation describe the role of individual atoms and associations of small numbers of atoms during the earliest stages of film formation. An important advance in the atomistic approach to nucleation was the theory proposed by Walton et al.(Ref. 21) which treated clusters as macromolecules and applied concepts of statistical mechanics in describing them. They introduced the critical dissociation energy, E_i, defined to be that required to disintegrate a critical cluster containing i atoms into i separate adatoms. The critical concentration of clusters per unit area of size i, N_i, is then given by

     (7-34)

which expresses the chemical equilibrium between clusters and monomers. In this equation, E^*_i may be viewed as the negative of a cluster formation energy, n₀ is the total density of adsorption sites, and N₁ is the monomer density. The latter, by analogy to Eq. 7-27, is given by . Hence

     (7-35)

Lastly, the critical monomer supply rate is essentially given by the vapor impingement rate and the area over which adatoms are capable of diffusing before desorbing. Therefore, through Eq. 7-32,

     (7-36)

By combining Eqs. 7-34 to 7-36, the critical nucleation rate (cm⁻²-s⁻¹) emerges as

     (7-37)

This expression for the nucleation rate of small clusters has been central to subsequent theoretical treatments and in variant forms has also been used to test much experimental data. Unlike our previous expression for (Eq. 7-33), to which it should be compared, it has the advantage of expressing the nucleation rate in terms of measurable parameters rather than macroscopic quantities such as ΔG* γ, or θ; these quantities characteristic of capillarity theory are not known with certainty, nor are they easily estimated. Now, however, the uncertainties are in i* and E^*_i.

One of the important applications of this theory has been to the subject of epitaxy, where the crystallographic geometry of stable clusters has been related to different conditions of supersaturation and substrate temperature. In Fig. 7-14 experimental data for Ag on (100) NaCl are shown together with the atomistic model of evolution to stable clusters upon attachment of a single adatom. At high supersaturations and low temperatures the nucleation rate is frequently observed to depend on the square of the deposition rate, i.e.,, indicating that i* = 1. This means that a single adatom is in effect the critical nucleus. At higher temperatures two or three atom nuclei are critical and the stable clusters now assume the planar atomic pattern suggestive of (111) or (100) packing, respectively. Epitaxial films would then evolve over macroscopic dimensions provided the original nucleus orientation was preserved with subsequent deposition and cluster impingement.

Figure 7-14 Nucleation rate of Ag on (100) NaCl as a function of temperature. Data for three different deposition rates are plotted. Also shown are smallest stable epitaxial clusters corresponding to critical nuclei.

(From Ref. 21.)

A thermally activated nucleation rate whose activation energy is dependent on the size of the critical nucleus is predicted by Eq. 7-37. This suggests the existence of critical temperatures where the nucleus size and orientation may undergo change. For example, the temperature T_1→2 at which there is a transition from a one- to a two-atom nucleus is given by equating the respective nucleation rates . After substitution and some algebra,

     (7-38a)

or

     (7-38b)

Use can be made of Eq. 7-38b in analyzing the data shown in Fig. 7-13. The transition of polycrystalline Cu to single-crystal Cu films can be thought of as the onset of (111) epitaxy (i* = 2) from atomic nuclei (i* = 1).

An Arrhenius plot of the line of demarcation between mono- and polycrystal deposits yields an activation energy of E_des+ E₂ equal to 1.48 eV. At a deposition rate of 8.5 × 10¹⁴ atoms/cm²-s or 1 Å/s, and n₀ν = 6.9 × 10²⁷atoms/cm² -s, the epitaxial transition temperature is calculated to be 577 K. Equations similar to Eq. 7-38 can be derived for transitions between i* = 1 and i* = 3, or i* = 2 and i* = 3, etc.

It is instructive to compare these estimates of epitaxial growth temperatures with ones based on surface-diffusion considerations (Ref. 22). For the case of smooth layer-by-layer growth on high-symmetry crystal surfaces, adatoms must reach growth ledges by diffusional hopping before other islands nucleate. Atoms must, therefore, migrate a distance of some 100–1000 atoms, the typical terrace width on well-oriented surfaces during the monolayer growth time of ∼ 0.1−1 s. By Eq. 7-30 a surface diffusivity greater than ∼ 10⁻⁸ cm²/s is, therefore, required irrespective of the material deposited. This means a different critical epitaxial-growth temperature (T_E) depending on whether a semiconductor, metal, or alkali halide films is involved. In the Arrhenius plot of Fig. 7-15, D_S values are graphed vs T_M/T where T_M is the melting point. Similar relationships for lattice, grain-boundary, and dislocation diffusivities will be introduced in Chapter 11. At the critical value of D_S = 10⁻⁸ cm²/s, Fig. 7-15 indicates that T_E ∼ 0.5T_M, ∼ 0.3T_M, and 0.1T_M for layer growth on group IV semiconductors, metals, and alkali halides, respectively. These temperatures agree qualitatively with experimental observations.

Figure 7-15 Lattice (D_L) and surface diffusivities (D_S) as a function of T_M/T for metals, semiconductors, and alkali halides.

(From Ref. 22.)

7.4.4 KINETIC MODELS OF NUCLEATION

Microscopic approaches to the modeling of nucleation processes have stressed the kinetic behavior of atoms and clusters containing a small number of atoms. Rate equations similar to those describing the kinetics of chemical reactions are used to express the time-dependent change of cluster densities in terms of the processes which occur on the substrate surface. Because these theories and models are complex mathematically as well as physically, the discussion will stress the results without resorting to extensive development of derivations. It is appropriate to start with the fate of the mobile monomers. If coalescence is neglected, then

     (7-39)

The equation states that the time rate of change of the monomer density is given by their incidence rate, minus their desorption rate, minus the rate at which two monomers combine to form a dimer. This latter term follows second-order kinetics with a rate constant K₁. The last term represents the loss in monomer population due to their capture by larger clusters containing two or more atoms. There are similar equations describing the population of dimer, trimer, etc., clusters as they interact with monomers. The general form of the rate equation for clusters of size i is

     (7-40)

where the first term on the right expresses their increase by attachment of monomers to smaller i− 1 sized clusters, and the second term their decrease when they react with monomers to produce larger i+1 sized clusters. There are i of these coupled rate equations to contend with, each one of which is dependent on direct impingement from the vapor as well as desorption through their link back to Eq. 7-35. Inclusion of diffusion terms, i.e., d²N_i/dx², enables change in cluster shape to be accounted for. When cluster mobility and coalescence are also taken into account, a fairly complete chronology of nucleation events emerges, but at the expense of greatly added complexity.

Transient as well as steady-state (i.e., where dN_i dt = 0) solutions have been obtained for the foregoing rate equations for a variety of physical situations and for arbitrary values of i. They typically predict that N(t) increases with time, eventually saturating at the value N_s. In practice the stable nucleus density is observed to increase approximately linearly with deposition time, then saturate at values ranging from ∼ 10⁹ to 10¹² cm⁻² depending on deposition rate and substrate temperature (Ref. 19). With further deposition N(it) decreases as a result of coalescence. Furthermore, N_S is larger the lower the substrate temperature, as shown schematically in Fig. 7-16.

Figure 7-16 Schematic dependence of N(t) with time and substrate temperature. T₁>T₂>T₃>T₄.

(From Ref. 19.)

Venables (Ref. 23) has neatly summarized nucleation behavior for cases where iassumes any integer value. In general, the stable cluster density is given by

     (7-41)

where A is a calculable dimensionless constant dependent on the substrate coverage. As a result of such generalized equations, experimental data for N_S have been tested as a function of the substrate temperature and deposition rate. The parameters p and E have respectively yielded values for i* and the energies of desorption, diffusion, and cluster binding. Values for E_des and E_S have been extracted for a number of systems, and these are entered in Table 7-1. Because E_des > E_S, adatom motion is largely confined to the substrate plane rather than normal to it. Although the major application of the kinetic model has been to island growth, the theory is also capable of describing S-K growth.

Table 7-1 Nucleation Desorption and Surface Diffusion Energies^a

7.4.5 CLUSTER COALESCENCE AND DEPLETION

As previously noted, the density of stable nuclei increases with time up to some maximum level before decreasing because of coalescence phenomena. Growth and coalescence of nuclei are generally characterized by the following features:

Several mass transport mechanisms have been proposed to account for these coalescence phenomena, and they are discussed in turn.

7.4.5.1 Ostwald Ripening

Prior to coalescence there is a collection of islands of varied size, and with time the larger ones grow or “ripen” at the expense of the smaller ones. The desire to minimize surface free energy of the island structure is the driving force. To understand the process, consider two isolated islands of surface tension y and different size in close proximity as shown in Fig. 7-17a. For simplicity they are assumed to be spherical with radii r₁ and r₂, and therefore the free energy (G) of a given island is 4πr²_iγ (i = 1, 2). The island contains a number of atoms n_i given by 4πr³_i/3Ω, where Ω is the atomic volume. Defining the free energy per atom μ_i or chemical potential as dG/dn_i, after substitution,

Figure 7-17 Coalescence of islands due to (a) Ostwald ripening, (b) sintering, (c) cluster migration.

     (7-42)

In chemical thermodynamics the chemical potential is often associated with the so-called “escaping tendency” of atoms. Where μ is large the effective atomic concentration is large, forcing them to escape to where μ is small. This is the case just outside very small particles. Thus, if r₁ > r₂ as shown, μ₂ > μ₁, and atoms will diffuse along the substrate from particle 2 (shrinking it in size) to particle 1 which grows at the former’s expense. A mechanism has thus been established for coalescence without the necessity for islands to be in direct contact. In a multi-island array the kinetic details are complicated, but ripening serves to establish a quasi-steady-state island size distribution which changes with time. Ostwald ripening processes never reach equilibrium during film growth, and the theoretically predicted narrow distribution of crystallite sizes is generally not observed.

7.4.5.2 Sintering

Sintering is a coalescence mechanism involving islands in contact. It can be understood by referring to Fig. 7-18, depicting a time sequence of coalescence events between Au particles deposited on molybdenite (MoS₂) at 400°C and photographed within the TEM (Ref. 24). Within tenths of a second a neck forms between islands and then successively thickens as atoms are transported into the region. The driving force for neck growth is simply the natural tendency to reduce the total surface energy (or area) of the system. Since the algebraic magnitude of μ for atoms on the convex island surfaces (r > 0) exceeds that for atoms situated in the concave neck (r < 0), an effective concentration gradient between these regions develops.

Figure 7-18 Successive electron micrographs of Au deposited on molybdenite at 400°C illustrating island coalescence by sintering. (a) Arbitrary zero time, (b) 0.06 s, (c) 0.18s, (d) 0.50 s, (e) 1.06 s, (f) 6.18 s.

(From Ref. 24.)

This results in the observed mass transport into the neck. Variations in island surface curvature also give rise to local concentration differences that are alleviated by mass flow.

In the case of sintering or coalescence of contacting spheres of radius r (Fig. 7-17b), theoretical calculations in the metallurgical literature (Ref. 25) have shown that the sintering kinetics is given by

     (7-43)

Here X is the neck radius, A(T) is a temperature-dependent constant that varies with mass transport mechanism, n and m are constants, and t is the time. Of the several mechanisms available for mass transport in films, the two most likely ones involve diffusion either through the bulk or via the surface of the islands. For bulk diffusion n = 5, m = 2, whereas for surface diffusion n = 7, m = 3. Typical thumbnail calculations show that surface diffusion dominates sintering.

While surface energy and diffusion-controlled mass transport mechanisms undoubtedly influence liquid-like coalescence phenomena, sintering mechanisms are unable to explain the following:

1. Observed liquid-like coalescence of metals on substrates maintained at 77 K where atomic diffusion is expected to cease

2. Widely varying stabilities of irregularly shaped necks, channels, and islands possessing high curvatures at some points

3. The large range of times required to fill visually similar necks and channels

4. Enhanced coalescence in the presence of an applied electric field in the substrate plane

7.5.1 ELECTRON MICROSCOPY

The traditional tool, particularly for island growth studies, has been the transmission electron microscope (TEM). In this technique metals such as Ag and Au are deposited onto substrates of interest such as cleaved alkali-halide single crystals, e.g., LiF, NaCl, and KCl, in an ultrahigh-vacuum system for a given time at fixed . The deposit is then covered with carbon and the substrate is dissolved away, leaving a rigid carbon replica which retains the metal clusters in their original crystal orientation. To enhance the very early stages of nucleation the technique of decoration is practiced. Existing clusters are decorated with a lower melting-point metal such as Zn or Cd which does not condense in the absence of prior noble metal deposition. This technique renders visible otherwise invisible clusters which may contain just a few atoms, thus making possible more precise comparisons with theory.

A disadvantage of the foregoing postmortem step-by-step observations is that much useful information on intermediate stages of film nucleation and growth is lost. For this reason, in situ techniques within both scanning (SEM) and transmission electron microscopes, modified to contain deposition sources and heated substrates, have been developed, but usually at the expense of decreased resolution. A recent in situ TEM study by Ross (Ref. 29) of the CVD deposition of Ge on Si (001) substrates illustrates the power of such methods. The Ge was grown at 650°C from digermane (Ge₂H₆) gas at pressures between 10⁻⁸ and 10⁻⁶ torr while dark-field images were continuously recorded. Epitaxial nucleation was deemed to occur after strain contrast was first detected at a dose of 50 L (or 19 monolayers; see Section 2.2.3) of Ge₂H₆. Two island shapes–square-bottomed pyramids and steep domes–then grew in a bimodal size distribution. Within minutes at a growth rate of 0.1 to 10 monolayers/min the smaller pyramids disappeared, leaving a sole distribution of dome-shaped islands ∼ 60 nm in size. An Ostwald ripening–like mechanism was suggested to be responsible for the changes in nucleus density. Such observations have enabled a better understanding of the processing of “quantum dot” structures.

7.5.2 AUGER ELECTRON SPECTROSCOPY (AES)

This technique is based on the measurement of the energy and intensity of the Auger electron signal emitted from atoms located within some 5–10 ˚ of a surface that is excited by a beam of incident electrons. The subject of AES will be treated in greater detail in Chapter 10. Here it is sufficient to note that the Auger electron energies are specific to, or characteristic of, the atoms emitting them and thus serve to fingerprint elemental composition. The magnitude of the AES signal is related directly to the abundance of the adatoms in question. Consider now the deposit/substrate combinations corresponding to the three growth modes (Ref. 30). If the AES signal from the film surface of each is continuously monitored during deposition at a constant rate, it will have the coverage or time dependence shown schematically in Fig. 7-19, assuming a sticking coefficient of unity.

Figure 7-19 Schematic Auger signal currents as a function of time for the three growth modes: (a) island, (b) planar, (c) S-K. o = overlayer, s = substrate.

(From Ref. 30.)

In the case of island growth (Fig. 7-19a) the signal from the deposit atoms builds slowly while simultaneously that from the substrate atoms correspondingly falls. The interpretation of the AES signal in the case of layer growth (Fig. 7-19b) is more complicated. During growth of the initial monolayer, the Auger signal is proportional to the deposition rate and sticking coefficient of adatoms, as well as to the elemental detection sensitivity. For the second and succeeding monolayers the sticking coefficients change. This gives rise to slight deviations in slope of the AES signal each time a complete monolayer is deposited, and therefore the overall response is segmented as indicated. For S-K growth (Fig. 7-19c) the signal is ideally characterized by an initial linear increase up to one monolayer or sometimes a few monolayers. Then there is a sharp break and the Auger amplitude rises slowly as islands, covering a relatively small part of the substrate, emerge. It would be misleading to suggest that all AES data fit one of the categories in Fig. 7-19. Complications in interpretation arise from atomic contamination, diffusion and alloying between deposit and substrate, and transitions between two- and three-dimensional growth processes.

7.5.3 SOME RESULTS FOR METAL FILMS

The properties of metal particles and films on nonmetallic substrates have played an important role in optical coating and ceramic bonding technologies, and particularly in the performance and selectivity of catalysts (Ref. 31).

In addition, studies of the nucleation and growth of metals, especially the noble ones, on assorted alkali halide and oxide substrates have long provided the basis for understanding epitaxy and film formation processes. An appreciation of the scope of this past as well as present research activity on metal/substrate systems can be gained by noting that more than 300 references to the research literature refer just to epitaxial Au film/substrate combinations (Ref. 32). Other epitaxial metal-film/substrate systems have been comprehensively tabulated in the earlier literature (Ref. 33) together with deposition methods and variables. The sheer numbers and varieties of metals and substrates involved appears to point to the fact that epitaxy is a rather common phenomenon; island growth is involved in the overwhelming number of studies, but as a caveat the reader should see Sections 8.2.2 and 8.3.5. In the case of Au, epitaxial films have been deposited on metallic, covalent, and ionic substrate. Although the majority of substrate materials have cubic crystal structures, this is not an essential requirement for epitaxy, which occurs, for example, on hexagonal close-packed Zn as well as on monoclinic mica. The fact that epitaxy is possible between materials of different chemical bonding and crystal structure means that its origins are not simple. That a small difference in lattice constants between film and substrate is essential for epitaxy is apparently not true; small lattice mismatch is neither a necessary nor a sufficient condition for epitaxy. The lattice parameter of the metal can either be larger or smaller than that of the substrate. Having said this, it is also true that the defect density in these metal-film “island” epitaxial systems is very much larger than in the “planar” epitaxial semiconductor systems; in these very close lattice matching is maintained.

The following findings, now primarily of historical interest, briefly summarize numerous studies of nucleation and epitaxy of metal films deposited on ionic substrates (Ref. 34).

7.5.4 SCANNING PROBE MICROSCOPY STUDIES OF NUCLEATION

Saved for last, we briefly note several more recent nucleation studies performed with the scanning tunneling (STM) and atomic force microscopes (AFM). These techniques, capable of unprecedented levels of resolution, have dramatically enhanced our understanding of surfaces and the atomic processes that occur on them. In particular, height variations at atomic steps are routinely measured with a resolution of 1 Å by STM. This makes it possible to determine whether islands have grown in only one atomic layer at a time, or atop one another in three dimensions. We start by considering the research by Lagally and co-workers (Refs. 8, 35, 36) on the earliest stages of Si atom homoepitaxy on a (001) Si substrate surface.

7.5.4.1 Silicon on (001) Si

Earlier in Section 7.2.4.1 it was pointed out that the (001) Si surface exhibits a (2 × 1) reconstruction and contains dimer rows. These rotate 90° as successive terraces are accessed giving rise to two types of monatomic steps. When silicon atoms are deposited on such surfaces the new layer of adatom islands form dimer rows. Occasionally, individual adatoms or pairs of atoms are visible. In addition to adatoms there are ad-dimers, which form when two adatoms pair up. Theory and experiment reveal that ad-dimers can translate as well as rotate. Although they exist predominantly on the dimer tops initially, with time they tend to populate the more stable trough configurations between dimers. Deposition on three successive terraces of the surface is shown in Fig. 7-20, each with their adatom island chains (light). Measurement of the island densities from such images recorded at a series of substrate temperatures yields the Arrhenius plot of Fig. 7-21. Because wandering adatoms either find an existing terrace edge or create new islands on the terrace by combining with other adatoms, the density of islands under specified deposition conditions is a measure of the diffusion coefficient. Thus, the energy barrier for surface diffusion determined from Fig. 7-21 is 0.65 eV, the value entered in Table 7-1. Direct STM observation has also revealed strongly anisotropic diffusion of Si adatoms. At 500 K adatom transport is 1000 times faster in the direction parallel to dimer rows than perpendicular to them.

Figure 7-20 Scanning tunneling microscope image of Si deposition on three successive terraces. Field of view is 300 Å × 300 Å.

(From Ref. 35.)

Figure 7-21 Arrhenius plot of island density vs the reciprocal of the substrate temperature.

(From Ref. 35.)

7.5.4.2 Metals on Silicon

In a similar vein, Neddermeyer (Ref. 10) has reviewed studies of the initial stages of metal (Ag, Cu, Pd) film growth on (111) Si substrates. Atoms on this close-packed Si plane undergoes a very complex structural reconstruction that was mentioned in Section 7.2.4.2. In the case of silver, when only one-third of a monolayer deposits on this surface at 90°C, the STM image depicted in Fig. 7-22a is obtained. Individual adatoms of Ag in the form of a ringlike triangular entity denoted by A and B are seen to sit above the otherwise undisturbed substrate atoms. When additional Ag atoms attach to the initial nuclei, the larger faceted clusters C and D form.

Figure 7-22 Scanning tunneling microscope images of deposited metal clusters. (a) Ag nuclei on (111) Si (Ref. 10). (b) A 0.16 monolayer deposit of Ag on (111) Au.

(From Ref. 37.) (c) The 11th monolayer of Cu electrodeposited on Au. (From Ref. 38.) (Reprinted with the permission of the authors.)

7.5.4.4 W_xN on SiO₂

In a last example, let us consider the use of the atomic force microscope (AFM) as a tool in studying film formation of W_x N on amorphous SiO₂. The function of this instrument is described in Section 10.3.5.4. Here we simply note that the AFM also traces surface topography, perhaps not with the spatial resolution of the STM, by detecting the van der Waals force interaction between tip and surface. In a study by Li et al.(Ref. 39), nucleation and growth of W_x N films by plasma-enhanced CVD at a deposition temperature of 340°C and rate of 1 nm/s was recorded during the first few seconds. The film topography is shown in Figs. 7-23a, b, c accom panied by information on film chemistry as revealed by Auger spectroscopy (Section 10.4.4). After 1 s there was no detectable W or N Auger signal, so this period may be taken as the deposition incubation time. A W_x N layer formed after 2 s consisting of 320 grains in an area measuring 500 nm × 500 nm, where each grain was ∼ 25 nm in diameter. Cluster coalescence occurred within 3 s, leaving 180 grains within the same area, where the size of each grain was now ∼ 45 nm. Clearly, the lateral grain dimensions are much larger than the nominal film thickness.

Figure 7-23 Atomic force microscope images of W_xN on SiO₂ after (a) 1s, (b) 2 s, and (c) 3 s. The respective surface rms roughnesses are 0.13 nm (the same as for the original SiO₂), 0.50 nm, and 0.48 nm. Each division represents a length of 100 nm along the horizontal axis and a height of 3 nm on the vertical axis. Corresponding Auger spectra appear in the inset.

(From Ref. 39. Reprinted with the permission of the authors.)