Recent theoretical developments in epitaxy

Materials Chemistry and Physics, 36 (1993) l-30 1

Invited Review

Recent theoretical developments in epitq

Ivan Markov” Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu (Taiwan, ROC)

Abstract

Recent developments in the theory of the mechanism of growth of thin epitaxial films and the models of epitaxial interfaces to account for the non-Hookean effects in atomic interactions are reviewed. It is shown that the growth mode criterion derived by Bauer in terms of specific free surface energies is in fact equivalent to the one based on the thickness dependence of the chemical potential in ultrathin films. The latter is more general and allows direct calculation of the critical temperature for the 2D-3D transition. Frank and van der Merwe’s 1D model of epitaxial interfaces is refined to account for the anharmonicity and nonconvexity of the real interatomic forces. Comparison shows that the available experimental data concerning the dislocation structure of the interface and the equilibrium thickness for pseudomorphous growth are in qualitative agreement with the consequences of the model. New properties of the interface, including distortion of the chemical bonds (alternation of long, weak and short, strong bonds) and the existence of multisolitons (coupled soliton-antisoliton solutions), are predicted in expanded epilayers.

1. Introduction

Since the time Royer [l] formulated his famous rules, the phenomenon for which he coined the term ‘epitaxy’ has become the foundation of powerful technologies for the fabrication of numerous novel devices [24]. Microelectronic and optoelectronic devices are prepared by epitaxial growth of materials varying from such ‘simple’ ones as elementary semiconductors (Si, Ge) and binary compounds and alloys (GaAs, Ge,Si,_,) to ternary and quaternary alloys such as In,Ga,_,As and InXGal -&,,P, _Y. By varying the composition (X and y) one can smoothly change the crystallographic parameters, e.g., the lattice parameter and in turn the lattice misfit, and the physical properties, e.g., the width of the forbidden energy gap in semiconductors. That is why numerous review papers and monographs [5-161 have been devoted to various aspects of the growth and characterization of epitaxial films.

Epitaxy is such a complex phenomenon that a unified and noneclectic theory of epitaxy is unlikely. The com- plexity arises from the variety of aspects of epitaxy and

*Permanent address: Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1040 Sofia, Bulgaria.

the interdependence of the large number of factors that affect them. Thus the nature and the strength of the chemical bonds in both materials and across the interface, on the one hand, and the lattice misfit, on the other, determine the structure and the energy of the boundary between two misfitting crystals. In turn, the latter determine the epitaxial orientation, the defect structure of the interface and its properties with respect to device application, the equilibrium critical thickness for pseudomorphous growth, the residual strain in the epilayers, the uniformity of the film thickness, the mechanism of growth of the epitaxial films and in turn the surface morphology of the growing films, etc. [ 13-161. The morphology of the growing films in turn affects the generation of misfit dislocations at the interface and the critical thickness for pseudomorphous growth. All of the above are of the utmost technological im- portance, as they affect the performance of the corresponding devices.

There are essentially two models of epitaxial interfaces that are commonly used. First, there is the one-dimensional misfit dislocation model of Frank and van der Merwe [18] valid for thin epilayers and developed further by van der Merwe [19, 201 (for a review see ref. 17) for the case of the interface between two semi-

0254-0584/93/$24.00 0 1993 - Elsevier Sequoia. All rights reserved

infinite crystals. Second, there is the dislocation model of Matthews [Zl] based on the Volterra concept of dislocations [22]. In both models the two crystals (substrate and deposit) are considered as elastic continua. As a consequence the properties of the interface do not depend on the sign of the lattice misfit, i.e., whether the epilayer is compressed or expanded.

A marked asymmetry in the critical thickness for pseudomorphous growth of molecular-beam epitaxy (MBE) grown In,Ga,_& [23, 241 and liquid-phase epitaxy (LPE) grown In,Ga,_&,,P, -Y [25] on InP(OO1) with respect to the sign of the misfit has recently been established. The critical thickness of the expanded epilayers turned out to be considerably greater than that of the compressed ones. In addition, Chen et al. [26] observed a partial pseudomorphism, or pseudomorphism in one direction, in the case of epitaxial growth of MoSi, on Si(ll1). The lattice misfit in two directions under an angle of 60” had the values 2.21% and -2.68%. Instead of a cross-grid of misfit dislocations, which is expected on the basis of the harmonic models, an array of parallel dislocation lines was observed even though the absolute value of the negative misfit was larger than the value of the positive misfit. This asymmetry of the properties of the epilayers can clearly be attributed to the anharmonicity of the interatomic bonding in the overgrowth.

The 1D model of Frank and van der Merwe [lS] has recently been extended to account for the influence of the anharmonicity [27,28] and nonconvexity [29,30] of the pairwise interatomic potentials on the equilibrium structure and energy of epitaxial interfaces. It was shown that the asymmetry of the interatomic bonding in the overgrowth leads to a split of all properties of the epitaxial interfaces, including the energy, mean dislocation density, stability limits and in turn the critical thickness for pseudomorphous growth, etc., with respect to the misfit sign. The anharmonic model qualitatively explained the experimental observations mentioned above. The influence of the nonconvexity of pairwise potentials beyond the inflection point turned out to be much more. complex. First, the nonconvex pairwise potentials display a maximal force between the atoms at the injection point, which can be considered as the theoretical tensile strength of the material. If a force greater than the maximal force is applied to a particular bond, the bond breaks and the chain loses its integrity. This explains the formation of cracks in compressed overlayers in which the bonds in the cores of the misfit dislocations are expanded. Second, the nonconvexity beyond the inflection point leads to distortion of the consecutive chemical bonds, consisting of alternating long, weak and short, strong bonds [31, 321. The distortion of the chemical bonds in turn affects the behaviour of the stability limits and the accommodation

of the lattice misfit by misfit dislocations (MDs) and homogeneous strain (HS). One consequence of the distortion of bonds that is still not completely understood is the existence of multisolitons (or coupled soliton-antisoliton solutions) in expanded overlayers [30]. The latter could be responsible for the formation of cracks in expanded epilayers.

The thickness uniformity of the growing fihns determines the smoothness of the growth front and the critical thickness for pseudomorphous growth, The latter are crucial for the performance of devices based on strained-layer superlattices [33]. The uniformity of the film thickness depends essentially on the mechanism of film growth. It was Bauer [34] who gave the clas- sification of the possible modes of growth. He found that if no interdiffusion, alloying or other changes in the surfaces take place, the mode of epitaxial growth should be determined by the interrelation of the specific free surface energies of the deposit (u), the substrate (uJ and the substrate-deposit, interface (pi). Layer-by- layer growth (or the Frank-van der Merwe mode), which is the most desirable for the preparation of smooth interfaces, should be expected when

a,>a+q

or, in other words, when the change in the surface energy Aa= cr+ ai - a, accompanying the deposition process is negative. Obviously, if the energy of the newly formed surfaces is smaller than the surface energy of the substrate, the formation of complete monolayers will be favoured energetically.

In the reverse case, when

as<U+Ui

or Acr> 0, the film will grow as isolated islands, so that large areas of the substrate with smaller surface energy will remain exposed. Island growth (or the VoI- mer-Weber mode) will take place. The combination of the above modes, namely, layer-by-layer growth fol- lowed by 3D islands, or the Stranski-Krastanov mechanism, takes place when Aho changes sign from negative to positive after some characteristic thickness. The latter is determined by the interrelation of the interatomic forces and the misfit energy per unit area [35]. Note that when Au=0 the film will grow as the bulk crystal by simultaneous growth of several monolayers.

Bauer’s criterion predicts the morphology of the films under near-to-equilibrium conditions, whereas the deposition usually occurs under conditions far from equilibrium. This is the reason a transition from layer-by- layer growth to either island growth or Stran- ski-Krastanov growth is often observed (far a review see ref. 16). Good examples are the deposition of Si on Geflll) and that of Ge on Si(ll1) 1361. In the first case, Si grows in the layer-by-layer mode at temperatures

3

lower than 600 “C and by separate islands above this temperature. In the second case, Ge grows in a layer- by-layer mode at temperatures lower than 500 “C, above this temperature the Ge film grows in a layer-by-layer mode up to a thickness of four monolayers and then 3D islands are formed on top of them. Thus a transition from Frank-van der Merwe mode to Volmer-Weber mode takes place in the first case and to Stran- ski-Krastanov mode in the second case.

A marked dependence of the mechanism of growth on the value of the natural misfit has been observed in the case of deposition of Ge,Si,_, alloy on Si(OO1) [37]. It was found that when the Ge content, x, exceeds 0.2 (so that the lattice mismatch is larger than 0.85%) the growth proceeds by formation of 3D islands. 2D growth, on the contrary, takes place when the Ge content is smaller than 0.2. The lower the Ge content, the thicker the Ge film that can be grown by successive monolayers pseudomorphous with the substrate. These observations were later confirmed by Bean et al. [38], who found that a pseudomorphous layer-by-layer growth can take place for alloys with a Ge content up to 0.5 and thicknesses as large as 0.25 pm.

It was later shown [lo, 39, 401 that the near-to- equilibrium morphology of the overgrowth can be predicted on the basis of the thickness dependence of the chemical potential. The layer-by-layer mode is to be expected when the chemical potential is an increasing function of the film thickness, i.e., when the derivative of the chemical potential p with respect to the film thickness n, dp(n)ldn, is positive [40]. In the reverse case, dF(n)l& <O, island mode should take place. The Stranski-Krastanov growth is determined by the chang- ing of the sign of the derivative dp(n)/dn from positive to negative after some critical film thickness. The latter is determined by the interplay between the difference of the cohesive and adhesive forces on the one hand and the misfit energy on the other.

We will show in Section 2 that both approaches are in fact equivalent. In order to do that we will start with the concept of the half-crystal or kink position introduced in the theory of crystal growth by Kossel [41] and Stranski [42] more than half a century ago. The second approach is, however, more general and allows direct calculation of the critical temperatures for transition from 2D (Frank-van der Merwe) to 3D (Volmer-Weber or Stranski-Krastanov) mode. This becomes obvious if we recollect that the difference of the chemical potentials is the driving force for the transport of material from sites with higher chemical potential to sites with lower chemical potential. The same approach naturally explains the appearance of a crosshatch pattern on the surface of the growing films [24].

In Section 3 we consider the influence of the anharmonicity and nonconvexity of the pairwise interatomic potentials on the properties of the classical one- dimensional (chain) model of Frank and van der Merwe [18]. In Section 4 we briefly discuss some aspects of the mutual dependence of the mechanism of growth and the consequences of the non-Hookean behaviour of the interatomic forces on the interface structure. Although the considerations will be directed mainly toward gaining a fundamental understanding, the technological impact of the results will also be discussed.

2. Mechanism of growth of thin epitaxial films

2.1. The concept of the half-crystal position We consider for simplicity the cubic face of a Kossel

crystal containing one monoatomic step (Fig. 1). The step is defined as the boundary between some region of the surface and an adjacent region whose height differs by one interplanar spacing. Atoms can occupy different sites on this surface. They can be incorporated into the crystal face (site 1) or the step (site 2) into the corner (site 3), or adsorbed at the step (site 4) or on the crystal surface (site 5). Depending on their positions, the atoms are bound differently to the crystal surface. Thus an atom adsorbed on the crystal surface is bound by one bond to the crystal and has five unsaturated dangling bonds. An atom incorporated into the topmost crystal plane, on the contrary, has five of its bonds saturated and one unsaturated. The detachment of these atoms leads to a change in the number of unsaturated dangling bonds, or, in other words, in the surface energy. The only exception is the atom in position 3, which has an equal number of saturated and unsaturated bonds. Then no change in the surface energy will take place when the latter is detached from this particular position. As can be seen from the figure, an atom in this position is bound to a half-atomic row, a half-crystal plane and a half-crystal block. This is the

Fig. 1. Schematic view of the sites an atom can occupy on a

crystal surface. Site 1, atom embedded in the uppermost crystal

plane; site 2, atom embedded into the outermost atomic row of

the step; site 3, atom in a half-crystal or a kink position (an

atom in this position is bound to a half-atomic row, a half-crystal

plane and a half-crystal block); site 4, atom adsorbed at the step;

site 5, atom adsorbed on the crystal surface.

4

reason this position is called a half-crystal position [41, 421. It is also known in the literature as a kink site or a growth site [43]. By repetitive attachment or detachment of atoms to and from this position, the whole crystal (if it is large enough to exclude the size effects) can be built up or disintegrated into single atoms.

The amount of work (P~,~ required to detach an atom from a half-crystal position depends on the symmetry of the crystal lattice, but is always equal to the work required to break half of the bonds of an atom situated in the bulk of the crystal. Thus for a Kossel crystal

where &, & and & represent the amount of work necessary to separate two first, second and third neigh- bours, respectively.

If we denote by Z,, Z,, Z,, etc. the coordination numbers of the first, second, third, etc. coordination spheres in an arbitrary crystal lattice, then

(1)

where N denotes the number of coordination spheres. When a sufficiently large crystal is in equilibrium

with the ambient phase, the half-crystal position is statistically occupied and unoccupied with equal frequency [42]. This means that the probability of attachment of atoms from the ambient phase to the kink position is equal to the probability of their detachment. It follows that the equilibrium of the infinitely large crystal with the ambient phase is determined by the half-crystal position. In other words, it is namely the work of separation from half-crystal (kink) position that determines the equilibrium vapour pressure of an infinitely large crystal and in turn its chemical potential. Thus for simple crystals with monoatomic vapours [44, 451, the relation

pc=pu,+kT In P,

= - q1,z + kT ln[(2mkT)3RkT/h3] (2)

holds, where m is the atomic mass and h is the Planck constant.

It follows from eqn. (2) that at T=O the chemical potential is equal to the work of separation from the half-crystal position taken with negative sign. It is namely this property of the half-crystal position that makes it unique in the theory of crystal growth [43].

As seen from Fig. 1, we can write the expression for the work of separation from a half-crystal position in the following form:

91, = (PIat + (Pnor (3)

where plat denotes the lateral bonding with the half- crystal plane and half-atomic row and qnor denotes the normal bonding with the underlying half-crystal block. This division has two advantages. First, it reflects the properties of the particular crystal face. Let us consider for example the most close-packed faces, (111) and (loo), of an fee crystal. We will restrict ourselves to first-neighbour interactions. In order to detach an atom from a half-crystal position on the (111) face, we have to break three lateral bonds and three normal bonds, whereas on the (100) face we have to break two lateral bonds and four normal bonds. In both cases we break six bonds, but we could conclude that the (100) surface has a greater adsorption potential than the (111) face.

Another very important consequence of this division is connected with the epitaxial growth of thin films. In fact, if we replace the underlying crystal block by another block of different material and assume additivity of the bond energies, the lateral bonding will remain approximately the same. However, the normal bonding, or the bonding across the interface, will change. Let us consider now the case when the lower crystal block is of a different material [46]. Restricting ourselves to first-neighbour bonds (& = s,!J), the work of separation from a half-crystal position of a monolayer on the surface of a foreign crystal with a simple cubic lattice will be

(P;n=4D*at+(Pnor=2~+~‘=cPl,z-(~-~) (4)

where 4pln =3$ is the work of separation from a half- crystal position of a bulk Kossel crystal and +’ is the energy needed to break a bond between unlike atoms.

In the general case

rp;n = ‘Pi/z - (% - &I> (5)

where CJQ (= qnor) and ~2 (=&,,) are the desorption energies from the same and the foreign crystal surfaces, respectively.

Bearing in mind eqn. (2), eqn. (5) can be rewritten in the form

&cn=cLcn+((pd-&) (5’)

where $_ and pm are the chemical potentials of the overgrowth layer and bulk crystal, respectively.

It is immediately seen that when (pd< 92, p’_ <pm and the equilibrium vapour pressure of the first monolayer on the foreign substrate must be smaller than the equilibrium vapour pressure of the bulk crystal, i.e., Pb. (1) <P,. Then at least one monolayer can be deposited at any vapour pressure higher than P’_(l). This means that deposition will take place even when P),(l) <P<P,, i.e., at undersaturation with respect to the bulk crystal [46]. In the opposite case (c,IJ~> &), ~6. >p, and P’_(l)>P,. This means that deposition requires the existence of a supersaturation in the system.

The atoms of the second monolayer feel the energetic influence of the substrate more weakly, and the latter will have a negligible effect on the atoms of the third monolayer [46].

2.2. Specific eneqg of the inte$ace between misfitting crystals

The substrate and deposit crystals differ not only in their lattices and lattice parameters, but also in the nature and strength of the chemical bonds. In the case of zero misfit, the quantity that properly gives the catalytic potency of the substrate, or in other words the energetic influence of the substrate on the film growth, is the specific adhesion energy /3, which is determined by the specific surface energies a, and a, of both crystals A and B and the specific interfacial energy ai through the relation of DuprC [lo, 471:

ai=U*+U,-p (6)

The latter can easily be derived. We perform the following imaginary experiment [lo]. We consider two semi-infinite crystals, A (substrate) and B (deposit), with equal dimensions (Fig. 2). We cleave them reversibly and isothermally and produce two surfaces of A and two surfaces of B, each one with area 2. In doing so, we expend energies U, and U,,. We then put the two halves of A in contact with the two halves of B and produce two interfacial boundaries AB, each one with area C. The work gained is -2U,. The excess energy of the boundary AB required to balance the energy change accompanying the above process is 2Ui. Thus we have

2ui = u, + u,, - 2u,, (7)

Clearly, when the two crystals are indistinguishable from each other, so that U,= U,,= UAB, the excess energy Ui = 0.

We use the definition of the specific surface energy, (T= U/2X, and define the specific interfacial energy a, as the excess energy of the boundary per unit area (a,= UJX) and th e specific adhesion energy p as the energy needed to disjoin two different crystals per unit

Fig. 2. The derivation of the relation of

semi-infinite crystals. DuprC in the case of

5

area (p=U,/X). Then eqn. (6) results. Note that p accounts for the binding between the two crystals and does not depend on the lattice misfit.

The interfacial energy q in this case accounts only for the difference in the nature and strength of the chemical bonds of both crystals. We assume further that the two infinitely large crystals A and B have different lattice parameters a and b. We cleave both crystals reversibly and isothermally as before and produce two surfaces of A and two surfaces of B (Fig. 2). In doing that we expend energies U, and U,, as before. We then strain uniformly the two halves of one of the crystals, say B, to match exactly the lattice parameter of the other, A, and put the halves of A and B in contact. Assuming that the lateral homogeneous strain does not affect the bonding across the interface, we gain the same energy - 2U,. After that we allow the bicrystal systems to relax so that MDs are introduced at the interface. The energy of HS is regained completely (the two crystal halves are semi-infinite), but an energy E, of a cross-grid of MDs is introduced in any one interface [17, 191. The energy balance now reads [16]

2ui=u,+u,,--2u,-+2E, (8)

Then

a*=u*+a,-/3+~~=u,+~~

or

(9)

a*=u,+u,-p* (10)

where Ed =E,/X is the misfit dislocation energy per unit area and the asterisks indicate that the corresponding quantities refer to misfitting crystals.

The specific adhesion energy between misfitting crystals, p*, is now given by the difference

p*=p-gd (11)

As seen, the dislocation energy appears as a decrement to the binding energy between both crystals. On the other hand, the dislocation energy appears as an increment to the specific interfacial energy, which is due to the lattice misfit. The remaining part, ui, is due to the different nature and strength of the chemical bonds and does not depend on the misfit.

We can repeat this process assuming now that crystal B is not semi-infinitely thick, but thinner than the double critical thickness 2.!, for pseudomorphous growth [17,21] (Fig. 3). We again cleave both crystals reversibly and isothermally, strain uniformly the halves of B to match exactly the lattice parameter of A, and put the halves of A and B in contact as before. Carrying out this operation we strain the free surfaces of B and change the specific surface energy. We assume that this change is much smaller than the work done to strain the crystals and neglect it. Then we allow the

6

Fig. 3. The derivation of the relation of Dupr6 in the case of a thin film on a semi-infinite crystal.

bicrystal systems to relax. MDs are not introduced at the interface, as the half-thickness of B is smaller than the equilibrium critical thickness for pseudomorphous growth and the pseudomorphous film is stable. The energy balance now reads

where 8, is the strain energy stored in crystal B per unit area of the interface function of the lattice misfit,

f* Finally, in the general case (crystal B is again thin,

but thicker than the double critical thickness 2& for pseudomorphous growth) only part of the HS is regained and MDs are introduced at the interface, but their density is partially reduced owing to the residual strain. Then [48]

and

p* =P-gecf,) -8dcf-fe)

(12)

(13)

where the strain energy Z=cfe) and the dislocation energy Z?&-fe) depend on the HS fe and the mean dislocation density f -fe, respectively.

2.3. Thickness dependence of the chemical potential As discussed in Section 2.1, the chemical potentials

of the first deposited layers differ from the chemical potential of the bulk deposit crystal, pm. First, the bonding with the substrate differs from that with the same crystal, and second, the lattice misfit leads to the appearance of HS and/or MDs. Elastically strained crystals have higher chemical potentials and hence the HS energy and the average value of the periodic strain energy due to the MDs contribute to the chemical potential of the film. On the other hand, the atom displacements affect the interaction across the interface and again lead to an increase in the chemical potentials of the first few layers deposited on the foreign substrate.

Bearing in mind eqns. (7) and (8), we combine eqns. (5’) and (13) and obtain, for the chemical potential

of the nth monolayer deposited on the foreign substrate,

p(n)=p_+a*[u+aT(n)-a,] (14)

where a* is the area occupied by an atom and Aa= o+ a*(n) - a, is the change in the surface energy connected with the deposition. The latter multiplied by a* is exactly equal to the chemical potential of the nth monolayer of the deposit relative to the bulk chemical potential.

It becomes obvious that Bauer’s criterion of the growth mode is completely equivalent to Ap(n) > 0 for island growth and Ap(n) ~0 for either layer-by-layer growth or Stranski-Krastanov growth, where Ap(n) = p.(n) -p_,. It should be pointed out, however, that the variation of the chemical potential with film thickness naturally includes in itself the thickness variation of the elastic strains due to the lattice misfit. In this respect the variation of the chemical potential is more general and suitable for calculation of the temperature of transition from 2D to 3D growth mode, as will be shown below.

Substituting a* from eqn. (12) in eqn. (14) gives, in terms of works of separation [48],

p(n) = pLm + 1% - &(n) + Ed(n) + 4n)l (1% where Ed(n) =a*ZY,(n) and ee(n) =a%Jn) are now the MD and HS energies per atom.

It is clear that when the substrate is of the same material as the deposit, rp,= ~2, fd=fe =0, l d= E,=O and p(n) = pm. It follows that the difference between crystal growth and epitaxial growth is clearly of a thermodynamic nature. We could even define the different kinds of epitaxial growth on the basis of eqn. (15). Thus if the main contribution to p(n) comes from the difference in bonding, qd - &(n), or in other words from the nature and strength of the chemical bonds in both partners, we have the case of heteroepitaxy. When the main contribution to p(n) comes from the misfit energy, the bonding in both crystals remaining essentially the same, we consider that to be homo- epitaxial growth. Finally, when p(n) = pL, we cannot speak of epitaxy at all.

rp:, can be either greater’ or smaller than qd, and hence the term in brackets can be either positive or negative and p(n) can be either greater or smaller than pm. In order to follow the p(n) dependence we should consider the thickness dependence of the quantities involved in eqn. (15).

The adhesion energy per atom, & accounts only for the atomic interaction across the interface in the absence of misfit. For short-range interactions it changes rapidly with film thickness, going to (Pd from above or from below (Fig. 4). The energetic influence of the substrate on atoms of the second monolayer will be very weak and can be neglected except in some extreme cases.

/’ NUMBER OF MLs /’

d

Fig. 4. Dependence of the energy of desorption of an atom from

an unlike substrate on the distance from the interface, measured

in number of monolayers.

FILM THICKNESS

Fig. 5. Dependence of the energy of the homogeneous strain of

the uppermost monolayer (curve 1) and the periodic strain energy

associated with the misfit dislocations stored beyond a thickness

h (curve 2) on the film thickness. tc: equilibrium critical thickness

for pseudomorphous growth; p/2: half spacing between the misfit

dislocations.

The thickness up to which E,, and E, contribute to the chemical potential depends in a complicated way on the lattice misfit [17]. The periodic strains due to arrays of MDs attenuate rapidly with the distance from the interface and practically vanish beyond a distance equal to half of the misfit dislocation spacing p (Fig. 5) [17]. Hence, cd =0 for t >p/2.

The HS energy E, is a parabolic function of the HS fe and a linear function of the thickness t. This means that every subsequent monolayer is strained to the same extent as the previous one up to the critical thickness t, for pseudomorphous growth. Above t,, fe and in turn E, rapidly decreases and can be neglected. Hence, we can simplify our considerations by assuming that l d = 0 and E,=E,(~) at t<t, and E,=O at t>t,.

We consider now some typical cases. We assume first that c&, > Q, i.e., the adhesive forces are stronger than the cohesive ones. If the misfit is small enough, so that E~-=K~_ and t, is large, then [48]

If the energetic influence of the substrate is felt not only by the first monolayer but also by the atoms of the second, though much more weakly, one observes the behaviour shown in Fig. 6 by the filled circles.

Let us assume now that rp: > (Pa as before, but the misfit is large, say, f4<f<fs, where f4 and f5 are the critical misfits at which the equilibrium critical thickness for pseudomorphous growth is exactly equal to 4 and 5 monolayers [16]. The energy of the HS is no longer negligible compared with pCLm. Then [48]

The latter p(n) dependence is shown in Fig. 6 by the semifilled circles. The gradual decrease of p(n) beyond the fourth monolayer reflects the decrease of the mean energy per atom of the periodic elastic strain due to the MDs with film thickness.

We assume further that & < 40~. If the misfit is small enough, so that t, is large, then

~(l)=~~+(Pd-~~+Ee(f)>~~

&!<n<t,/b)=~.,+c,>~m

Beyond t,, E, vanishes and MDs are introduced at the interface, so that E, is replaced by l d.

If the misfit is large and in the extreme case larger than fi, e,=O from the very beginning and

~(l)=&+(Pd-&+Ed>I-Lcc

p(2<n<p/2b)=pm +cd(n)>p~,

p(n >p/B) = pm

We consider the last case in more detail. The p(n)

dependence is decreasing, thus reflecting the decrease with film thickness of the periodic strain connected

NUMBER OF MLr

i (

Fig. 6. Schematic representation of the thickness dependence of

the chemical potential in ultrathin films: (0) Frank-van der

Merwe growth; (a) Stranski-Krastanov growth; (0) Volmer- Weber growth.

8

with the MDs (see Fig. 5). Then every subsequent monolayer will be on average less strained, will have a chemical potential smaller than that of the previous one and will start to form before the completion of the latter. In such a case the formation of islands thicker than one monolayer is thermodynamically favoured and their equilibrium with the ambient phase will be realized through a half-crystal position of multilayer height (Fig. 7). Then the chemical potential of the film consisting of several atomic monolayers ~(1 + n) will be given by the mean value of the chemical potentials of the constituent monolayers [16, 491:

The mean chemical potential ~(1 +n) is plotted in Fig. 6 with open circles.

2.4. Thermodynamic criterion of the growth mode It follows that depending on the interrelation between

the adhesive and cohesive forces, on the one hand, and on the value of the misfit, resulting in an interplay between the energies of the MDs and HS, on the other, three different types of thickness dependence of the chemical potential can be distinguished [40, 48-j (Fig. 6):

(1) d&hr < 0 when 40; < (pd at any value of the misfit; (2) dp/dn > 0 when cp; > (Pi at small misfits; (3) d&&z >< 0 when 9; > 40~ at large misfits. Obviously, when d&+z is positive the completion of

the first monolayer before the start of the second one, of the second before the start of the third, etc., is thermodynamically favoured. Hence layer-by-layer growth is expected. In the opposite case, the formation of a second monolayer before the completion of the first one is thermodynamically favoured and formation of 3D islands should take place. The third case is obviously a combination of the first two cases. The growth will begin by formation of a few monolayers, each one nucleating after the completion of the previous one. Then 3D islands will nucleate on top of the monolayer deposit. It follows that the above inequalities represent the thermodynamic criterion for the mechanism of growth of thin epitaxial films [16, 401:

(1) d&hr <O - Volmer-Weber growth; (2) dp/& > 0 - Frank-van der Merwe growth; (3) dp/dn 2 0 - Stranski-Krastanov growth.

Fig. 7. A bilayer half-crystal position.

Once the chemical potential acquires its bulk value pm the epilayer will grow further by simultaneous growth of several monolayers [52-541.

2.5. Kinetics of growth of thin epitaxial films We are now in a position to study the growth of

thin epitaxial films, bearing in mind the above thickness dependence of the chemical potential. The latter predicts the equilibrium morphology of the deposit, whereas the deposition process is usually carried out under conditions far from equilibrium. So we have to study how the substrate temperature and the rate of deposition affect the mechanism of growth.

We consider the case of complete condensation, when all atoms arriving at the crystal surface join sites of growth before reevaporation. As in any case of growth of a defectless, atomically smooth crystal face, the atoms from the vapour phase strike the substrate and, after a period of thermal accommodation, randomly walk to give rise to 2D nuclei. The 2D nuclei grow further by attachment of adatoms diffusing to their edges on the substrate surface and on their exposed surface as well. An adatom population is formed on top of them (Fig. 8(a)) whose concentration n,(r) can be found by solving the difhtsion equation [16, 501:

d*n, b-1 + i Wr) R dr* rdr+D,=O (16)

subject to the boundary conditions n,(r= p) =nzl and (dn,l&),,,=O, where p is the island radius. In eqn.

SUBSTRATE f

Fig. 8. Subsequent stages of film growth and atom exchange between the kinks and the dilute adlayer: (a) the concentration of atoms adsorbed on top of the first monolayer island increases with island size, which leads to nucleation of 2D islands of the second monolayer; (b) surface transport from the edgesA,BICIDI to the edges A2BZC2D2 takes place when ~(2) <p(l); (c) surface transport transforms the layer configuration into a crystal with a height of two monolayers, which grows further by nucleation of islands of the third monolayer.

9

(16) R (cm-’ s-‘) is the atom arrival rate and D, = u2v exp( - c&kT) is the surface diffusion coefficient, v and (Pi,, being the vibrational frequency of the adatoms and the activation energy for surface diffusion, respectively. The solution reads

n,(r) =n:, + $j- (p”-r2) s

(17)

The quantity nzl is the concentration of adatoms in equilibrium with the island edges, given by [16]

G =nse exp CL(l)-&- c 1 kT (18)

and n,, is the adatom concentration on the surface of the same bulk deposit crystal [51]

nse=no exp[ - v] (19)

where n,, is the density of adsorption sites on the crystal surface and the difference qln - (Pi is the work necessary for the transfer of an atom from a kink position on the crystal surface.

Equation (17) shows a parabolic dependence of the adatom concentration on the distance from the island centre (Fig. 9), which displays a maximum

R II s, Inax =G+ 40, P2

just over the island centre. The increase in the island radius p leads to values of n,, max high enough to give rise to nuclei on top of the islands (Fig. 8(b)). Thus nuclei of the second monolayer appear before the completion of the first one. Once such nuclei are formed they grow initially at the expense of the atoms diffusing

Fig. 9. The profile of the adatom concentration (a) on the surface

of a monolayer island, and (b) on the terrace formed between

the edges of the lower and upper monolayer islands with radii

p, and p2, respectively, n:, and n; are the adatom concentrations

in equilibrium with the corresponding island edges.

to their edges on the terrace between the edges of the upper and lower islands. The adatom concentration on the terrace (Fig. 9(b)) is easily found by solving eqn. (16) subject to the boundary conditions n&J =n$ and n&) =&, where

nz2=nn,, exp [ 1 p(2~~pm

is the adatom concentration that is in equilibrium with the edges of the second monolayer islands [16].

The solution that is plotted in Fig. 9(b) reads

R n,(r)=n:,+ 40 pf-r’

s ( )

In J

- b+ & M-P:) - I s 1

0 PI

0

(21)

In @ Pl

where An, =nE, - r&. When ~(1) > p(2)(d&ln ~0) the adatom population

on top of the first monolayer islands is supersaturated with respect to the bulk deposit crystal. The thermodynamic driving force for nucleation to occur on top of the first monolayer islands should be greater than Ap=~(l)--~. On the other hand n:, >r& and surface transport from the edges of the lower island to the edges of the upper island will take place, whose driving force is given by AnzlA = (n:, -n&)/h, where A is the distance between the edges (Fig. 9(b)). Thus the upper island will grow at the expense of the lower island and after some time the edges of the upper island will catch up with the edges of the lower one to produce an island of double height (Fig. 8(c)). Hence, at temperatures high enough to facilitate surface transport, island growth will be observed. However, if the temperature is low enough, surface transport from edge to edge will be hindered and the first monolayer islands will grow laterally to coalesce and completely cover the substrate surface before significant growth on top of them can take place. Layer-like growth will occur owing to kinetic reasons. However, such films grown at low temperature are metastable. Upon heating they will break up and agglomerate into 3D islands. Note that the growth will not follow the true layer-by-layer mechanism (complete coverage of the substrate surface by one monolayer before the next one nucleates), as the thermodynamic driving force favours island growth.

In the opposite case, ~(1) <p(2) (d&ln >O) (Fig. 9(b)), the islands of the second monolayer will have a chemical potential higher than that of the lower islands and surface transport of atoms will occur from

10

their edges toward the edges of the lower islands. As a result they will decay. Thus layer-by-layer growth will be observed irrespective of the temperature.

Finally, when dp.ldn changes sign with film thickness the first monolayers will grow layer by layer for the reasons given above. Once a particular thickness (the so-called Stranski-Krastanov thickness) is reached such that the corresponding chemical potential is higher than pm, 3D islands will form and grow at high temperatures. Surface transport from the edges of more elastically strained islands to the edges of less strained or not at all strained islands will take place. As a result, the Stranski-Krastanov mechanism will be observed. At low temperatures the growth will proceed further by formation and growth of monolayers (successive or simultaneous). Again, if such low-temperature films are annealed at higher temperatures the material in excess of the first stable monolayers will break up and agglomerate into 3D islands.

It is important to note, once more, that true layer- by-layer growth takes place only when the chemical potential is an increasing function of the film thickness, i.e., d+ln > 0. At low temperatures and dp/dn < 0 film growth will proceed by simultaneous growth of several monolayers. This is in fact the well-known multilayer growth in the theory of crystal growth [52-541.

2.6. Critical temperature for transition from 20 to 30 growth mode

Thus, we have to expect a change of the mechanism of growth from layer-by-layer (or, rather, multilayer growth) to either Volmer-Weber or Stranski-Krastanov growth with increasing temperature. Our next task is to find the critical temperature for the transition to occur.

We consider the case when p1 > pz > p3 . . . so that island growth is expected under near-to-equilibrium conditions. The same is valid in the case of Stran- ski-Krastanov growth after the completion of the first stable adlayers. As discussed above, 2D nuclei of the second, third, etc. monolayers are formed on top of the first monolayer islands, which results in the formation of flat pyramids of growth, as shown in Figs. 8(b) and 9(b). As the chemical potential is a decreasing function of the monolayer number n, the surface transport will be directed from the lower to the upper steps. We make use of the solutions (17) and (21) of the diffusion equation (16), assuming rapid exchange of atoms between the steps and the dilute adlayers on the terraces. We calculate the rates of advance of the circular steps vu, = dp, ldt [40].

Thus for the first monolayer island we have

2An:D, - No p1 ln(pZlp3

where N, is the density of the growth pyramids formed by successive 2D nucleation per unit area of the substrate and An: = n:, -n&

In this expression the first term on the right-hand side, which contains the deposition rate R, is always positive, as the surface coverage rp:N, is smaller than unity before the coalescence and p1 >pz. The second term, which contains the equilibrium concentration difference h:, is also positive, as n:, >n& It follows that v1 can be either positive or negative, depending on the values of the deposition rate R and the difference An,“, which is a steep function of the temperature. In the extreme case of the absence of deposition (annealing at R=O), the first term on the right-hand side vanishes and v1 <O, thus reflecting the process of detachment and transport of atoms from the lower monolayer island edge to the edge of the upper island during high- temperature annealing. The same process takes place during deposition, but at higher temperature, where the negative term overcompensates for the positive one. If this occurs before the coalescence of the first monolayer islands, say, at surface coverage 0, = rp;N, G 0.5, island growth has to be expected. On decreasing the temperature, An: decreases, and under a given temperature the term containing An: has a negligible contribution to vl. The rate v1 is then just the same as in the case of deposition on the same substrate, i.e., in the case of the growth of a bulk crystal when

P(n)=P-. We have to solve now a set of differential equations

for the rates of advance of the steps. The latter can be written in terms of surface coverages O,= rp:N, (n = 1, 2, 3 . ..) as a function of a dimensionless time 0=Rt/n,, which is in fact the number of monolayers deposited in the form

dO, =I- h!f,+@,-0,

de ln(O,/@)

d@n M,-~+@,-~--@, M,+O,-O”+,

de - ln(O,_,lO,) - ln(O,lO,+,) (22)

d@, M&_I+ON-_l-ON -= de ln(O,_,/O,>

where the subscript N denotes the uppermost monolayer and the parameters

M = 4~DsWn:,n-n:,n+l) n R (23)

11

include all material quantities and the differences of the adatom concentrations, or in other words the differences of the chemical potentials (see eqns. (18) and

(20)). Numerical analysis of the system (22) shows that the

solutions for 0, - and hence the time evolution of the shape of the growth pyramids - are very sensitive to the values of M,. The latter are strongly increasing functions of the temperature and are inversely pro- portional to the atom arrival rate R. When the chemical potentials are independent of the layer number, or in other words IZ~ =n then M, =O. In this case there is no directed &fag transport between the steps, and the growth pyramids preserve their shape. This means that the epitaxial film will grow as the bulk crystal face following the 2D nucleation mechanism with simultaneous growth of several monolayers [52-541.

Let us consider the simplest case, that of the bilayer pyramid, shown in Fig. 9(b), assuming in addition that

G = n,,, i.e., ~(2) = pm. As shown in Fig. 10, when M, =0.25 is positive (p(l)>p,), 0, increases initially, displays a maximum 0, = 0.5 and decreases. The latter means that at some stage of growth the rate of advance of the first monolayer island dp,/df becomes negative, or, in other words, the lower island decays and the atoms feed the upper island. Then the edges of the latter catch up with the edges of the former and an island with double height results (Fig. 8(c)). The double steps propagate more slowly than the single steps 1521, and after some time the single steps of the third monolayer islands catch up with the edges of the double- height island, thus producing an island with triple height. As a result, island growth takes place, the kinetic criterion for it being

4+WCl -hJ , o 25

R , .

1.2

i 1.0 + -7..

(24)

0.0 0.0 0.2 0.4 0.8 0.8

NUMBER OF MONOLIYERS

Fig. 10. Dependence of the surface coverages of the first (curves

1 and 1’) and second (curves 2 and 2’) monolayers on the number

of monolayers deposited. Curves 1 and 2: M=0.25; curves 1’

and 2’: M= -1.5.

The physical meaning of this criterion becomes trans- parent if we write it in the form [16]

The numerator represents the total diffusion flux from the edge of the lower island to that of the upper island resulting from the difference of the equilibrium adatom concentrations. The denominator is equal to the flux of atoms joining one pyramid. Therefore, the criterion simply states that in order for island growth to take place, the diffusion flux from edge to edge should be equal to or larger than 25% of the total number of atoms joining the pyramid. The increase in the deposition rate R leads to an increase in the overall growth rate of the pyramid without affecting the diffusion flux, which is responsible for the transformation of the pyramid to a 3D island. The result is a transition to multilayer growth. The increase in the temperature has the opposite effect: it results in a faster surface transport, which in turn facilitates the 2D-3D transformation.

It is interesting to see what will happen when M, has a negative value, i.e., when ~(1) < pCLm. As seen in Fig. 10 the surface coverage of the second monolayer 0, (curve 2’) decreases, reflecting the fact that the surface transport is directed from the upper to the lower island edges. The lower islands (curve 1’) grow at the expense of the upper ones and completely cover the substrate (0, = 1). True layer-by-layer growth results.

Making use of eqns. (18) and (19) for the transition temperature T, from eqn. (24) one obtains [48]

Tt= (~l/2-(Pd)-[IL(1)-~ml+(Psd k ln( 16rvNJR) (25)

The main contribution to ~(1) at small misfits is the difference 4~~ - 4~; of the energies to separate an adatom from the foreign substrate and from its own bulk crystal face (eqn. (15)). In the case of transition from multilayer to island growth we can write eqn. (25) in the following form [48]:

T [((PI,2 - &) - (cpd - FL)] - E, + (Psd = t

k ln(l6rVNJR) (26)

where E,,, is the misfit energy, equal to either Ed or Em. Keeping in mind that the lateral bonds of an atom

in a kink position remain practically unchanged, P,,~ - (pd = (p;,2 - cp; and the energy difference in the square brackets is just equal to the energy & - (Pi for transfer of an atom from a kink position in the step of the first monolayer island to the dilute adlayer on top of it.

Equation (26) is valid for the transition from multilayer to island growth where the contribution of the interatomic forces across the interface to the chemical

12

potential is large. In the case of the transition from multilayer to Stranski-Krastanov growth, the 3D islands are formed on top of one or two stable adlayers of the same material and (Pi- & can be neglected. As- suming the monolayers are pseudomorphous with the substrate and the 3D islands are relaxed, eqn. (26) simplifies to

T = (%2-(Pd)-ci+%d t k ln(l6TvN,/R) (27)

Then surface transport will occur from edges of more elastically strained monolayer islands to the edges of less strained or not at all strained monolayer islands.

It should be noted that whereas qln is characteristic for the bulk material, (P,, and 9; depend on the crystallographic orientation of the substrate. It follows that the critical temperatures will be considerably higher for the (111) face of an f.c.c. crystal (qPd=3+, (p;=3$‘) than for the (100) face (40, = 4$, 4”: = 4$‘). This is indeed what was observed in the deposition of a series of f.c.c. metals on W(110) and W(100) [55-591 (for reviews see refs. 16 and 60).

We consider now in more detail the deposition of Ge on Si(ll1) and Si on Ge(ll1) [36]. Transition from layer-by-layer to Stranski-Krastanov growth is observed when Ge is deposited on Si(lll), the transition temperature being 500 “C. The enthalpy of evaporation of Ge is 89 500 cal mol-‘, so that q1/2- qDd = 44 750 cal mol-‘. We calculate the MD energy per unit area 8, between the homogeneously strained Ge layers and relaxed 3D islands using the theory of van der Merwe [17] with a shear modulus of 5.46~ 1O1l dyn cm-‘, Poisson ratio vGe= 0.2 [61] and a lattice misfit of 0.041 and find the value ZYd =320 erg cm-‘. Then l d -3.4~ 10-20 J per atom=4900 cal mol-‘. Making use of eqn. (27) with N, = 1 X lo9 cm-’ and R = 0.1 ML S -‘=7.2x 1013 cmp2 s-’ [36], and assuming qsd is negligible, gives T,=530 “C, in good agreement with experimental observations.

The transition from layer-by-layer to island growth in deposition of Si on Ge(ll1) [36] is more difficult to handle. We assume first that the shear modulus at the interface, Gi, has a value in between the shear moduli of Si and Ge, 6.41 X loll and 5.46 X 10” dyn cmp2, respectively, and accept the average value Gi= (GSiGGe)1’2 = 5.9 X 101’ dyn cmp2. Then, with Pois- son ratios v,, = 0.2 and vsi = 0.215 [61], we find Zd = 340 erg cm -2, E~ = 3.35 x 10p20 J per atom = 4850 cal mol-I. The same assumption is made for evaluating the adhesion energy f&. From (pd(Si) = A&(Si)/2 = 54 450 cal mol-’ and pd(Ge) = m,(Ge)/2=44 750 cal mol-l we find y&- [cp,(Si)cp,(Ge)]‘R= 49 360 cal mol-’ and q,,(Si) - &(Si/Ge) = 5100 cal mol-l. Neglecting the ac-

tivation energy for surface diffusion, we find T,= 675 “C, which is 75 degrees higher than the experimentally found value 600 “C [36].

We have to bear in mind, however, that the theoretically predicted values of T, have been underesti- mated, as the activation energies for surface diffusion have been neglected. In the case of diffusion of Si atoms on Si(lll), the value 1.3 eV has been reported [62, 631. In addition, Sakamoto et al. [64] found that the surface diffusion on a vicinal Si(ll1) surface is anisotropic. Another uncertainty in calculating the transition temperatures comes from application of the theory of van der Merwe [17] for calculation of the MD energy in the case of materials with covalent bonds, which are considered to be brittle and inflexible. Additional uncertainty comes from using the nearest-neighbour model for calculation of the desorption energies. It is thus surprising that irrespective of all the approximations made, the above theoretical treatment is in good semiquantitative agreement with the experimental data.

2.7. Crosshatch patterns The theoretical model described above readily gives

a qualitative explanation of the appearance of the so- called ‘crosshatch patterns’ [24]. The latter represents an array of parallel lines or a grid of two arrays of mutually perpendicular lines on the surface of the growing epilayer where the latter is thicker than the remaining part of the film. Detailed investigation of the phenomenon in the case of the growth of In,Ga, _& on InP(lOO) [24] showed that each hatch line corresponds to a MD line. Thus the crosshatch pattern appears only when the interface is resolved in a cross- grid of MDs, although a one-to-one correspondence between the hatch lines and the MD lines has never been found. In addition, crosshatch patterns have been observed on the surface of the films under both tensile and compressive stress. Crosshatch patterns have never been observed on the surface of pseudomorphous films. Bearing in mind the thermodynamic analysis of the near-to-equilibrium morphology of growing epilayers given above, it is easy to assume that the parts of the film that are just over the MD lines are elastically relaxed, whereas the film remains elastically strained in between the lines. It follows that the chemical potential of the film over the dislocation lines is lower in comparison with that in the regions between the MD lines (Fig. 11) and the variation of I_L is just given by the energy of the HS Ed. Then surface transport of adatoms from regions with enhanced chemical potential to regions with lower chemical potential (denoted by the arrows in Fig. 11) takes place, just as in the case shown in Fig. 8(b). The parts over the dislocation lines grow thicker than the remaining parts of the film and the crosshatch pattern results. At low temperatures surface transport is inhibited and the appearance of

13

SUBSTRATE

Fig. 11. A schematic cross-sectional view of a crosshatch pattern. The upper curve illustrates a possibIe variation of the chemical potential of the crystal surface due to nonuniform distribution of misfit strain.

-b-

-a-

Fig. 12. The one-dimensional model of Frank and van der Merwe [18] representing a chain (monolayer) of atoms in a periodic potential field (substrate). As b > a, 11 atoms are distributed over 12 potential troughs, thus forming a misfit dislocation.

crosshatch patterns should be suppressed. In other words, the film surface should remain smooth.

3. Non-H~kean effects in epitaxial interfaces

The 1D model of epitaxy of Frank and van der Merwe [18], which deals with a linear chain of atoms subject to an external periodic potential exerted by a rigid substrate (Fig. 12>, has recently gained prominence in various fields, the common feature of which is the competing periodicities. Thus it provided the grounds of the theory of commensurate-incommensurate phase transitions in physisorbed layers [65] and in layered compounds [66f, the alignment of cholesteric liquid crystals in a magnetic field 1671, etc. (for a review see ref. 68). Although simple, the model qualitatively gives all the main properties of the epitaxial interfaces cor- rectly. That is why numerous attempts to relax some of the restrictions imposed by the authors have been initiated [69, 701. One of the basic restrictions of the model o~ginally adopted by the authors, which makes it applicable for small lattice misfits only, is the purely elastic (Hookean) interactions between neighbouring atoms as a substitute for the real interatomic forces. As mentioned in the Introduction, the replacement of the harmonic potential by more realistic anharmonic and nonconvex pai~ise potentials 127-301 made possible the explanation of available experimental data 123-261 and predicted new properties of the epitaxial interfaces. That is why we will consider the effects of anharmonicity and nonconvexity of the real atomic interactions in more detail.

In the original 1D model of Frank and van der Merwe [18] the overgrowth is simulated by a chain of atoms

connected by purely elastic (Hookean) springs as a substitute for the real interatomic forces. The springs are characterized by their natural length b and force constant E (Fig. 12). The chain is subject to an external periodic potential field characterized by a period a and the amplitude W exerted by a rigid substrate. The assumption of substrate rigidity could be considered to reflect the real situation in the case of thin-enough overgrowth (consisting of no more than a few monolayers). In the case of a thick deposit, this assumption is no longer valid and elastic strains in both substrate and deposit should be allowed.

The potential energy of the classical 1D chain consisting of N atoms is given by f18]

N-2 N-l

E= c. ~(x,.,., --K-b) + c WW”> ?l=O n-0

where Y(x) and W(x) are the interatomic and interfacial potentials, respectively. X, is the displacement of the nth particle with respect to some reference position. The expression X,,, -X,-b represents the strain of the nth spring (chemical bond), and thus the first sum gives the strain energy of the system. The second term gives the increment of the energy of interaction with the substrate when the atoms are displaced from their ideal positions in the bottoms of the potential troughs.

Two forces act on each atom: the force exerted by the neighbouring atoms, and the force exerted by the substrate. The first force tends to preserve the natural atom spacing b between the atoms, whereas the second one tends to place all the atoms at the bottoms of the corresponding potential troughs of the substrate and to space them at a distance a. As a result of the competition between the two forces, the atoms are spaced in general at some compromise distance 6 in between b and LI. when b-u, the natural misfit is accommodated by HS and the overgrowth is pseudomorphous with the substrate. In the other extreme, 6=b, the deposit preserves on average its own atom spacing and the natural misfit is accommodated entirely by MDs. It follows that in the intermediate case, a <b< 6, part of the natural misfit

will be accommodated by MDs

and the remaining part

6-b

fe=,= (31)

14

will be accommodated by HS. In other words, the natural misfit appears in the general case as a sum of the HS and the periodic strain due to the MDs, i.e.,

f =fLi + lfel (32)

3.1. Interatomic potentials Anharmonic and nonconvex interactions are common

in solid state physics. The pairwise potentials of Morse and Lennard-Jones with a simple analytical form are the most common choices [71].

The potential of Morse [72]

V(r)=&{ [ l-exp( -yy)r-l} (33)

where V0 is the energy of dissociation, r,, is the equilibrium atom spacing and y is a constant that governs the range of action of the interatomic forces, was originally suggested for evaluation of the vibrational energy levels in diatomic molecules. By varying y we shift both the repulsive and attractive branches in opposite directions, so that the degree of anharmonicity remains practically unchanged. Girifalco and Weiser [73] adjusted the constants of the Morse potential to fit the lattice parameters, the cohesive energies and the elastic properties of a series of metals and found a value for y varying around 4.

The potential of Morse does not behave well at small and large atom spacings. At r+ 0, the potential does not go to infinity but has a finite (although very large) value. The exponential dependence is not believed to describe well the atom attraction at r>r,, [71]. In this respect the inverse power Mie potential

is much more flexible than the Morse potential. The repulsive and the attractive branches are governed by two independent parameters, m and n (m>n). The Mie potential with m = 12 and rz =6, which is known as the Lennard-Jones potential [74], describes satis- factorily the properties of the noble gases.

A generalized Morse potential

V(r) = V, v e (

-&-nd _ P - ,-dr-m)

P--V P--V )

(35)

has recently been suggested [29]. It has all the short- comings of the original Morse potential, except that the repulsive and the attractive branches are governed by two independent parameters p and v (pL> v). An advantage of both Morse potentials, particularly for solving interface problems, is that they are expressed in terms of strains r-r,, which makes the mathematical formulation of the problem and the calculation of the

strains, stresses and associated strain energy easier. If we put p=2-y/r0 and u= y/r,, in eqn. (35), the latter turns into the classical Morse potential. It is worth noting that the 6-12 Lennard-Jones potential is practically indistinguishable from the generalized Morse potential (eqn. (35)) with p = 18 and V= 4. The potential expressed by eqn. (35) with p= 4 and v=3 is plotted in Fig. 13.

The pairwise potentials given above have two fundamental properties. First, they are anharmonic in the sense that the repulsive branch is steeper than the attractive one, and second, they have an inflection point ri beyond which they become nonconvex. In order to distinguish the influence of the anharmonicity from that of the nonconvexity, one can use the Toda potential

[751

V(r)=V, (

$ e -P’r-d++(r-rO)_ ‘y -1 P 1

which is plotted in Fig. 13 with IX= 2 and p= 6. By varying cx and p but keeping their product constant, we go smoothly from the harmonic approximation

( LY+ 03, p+ 0, @=const.) to the hard-sphere limit (a + 0, p + CO, (up = const.). The potential of Toda has no inflection point (or has an inflection point at infinity) and can be used to study the effect of anharmonicity in its pure form [27, 28, 761. It is immediately seen that expanding the second exponential term of the generalized Morse potential (eqn. (35)) in a Taylor series to the linear term results in the Toda potential with CY= &(p-- V) and p= I_L.

Expanding any of the above potentials in a Taylor series to the parabolic term gives the harmonic ap-

Fig. 13. The generalized Morse potential (eqn. (35)) (referred to as a ‘real’ potential) with cr. = 4, V= 3 and Va= 1, and the Toda potential (eqn. (36)) with a=2 and p=6. The parameters are chosen in such a way that the repulsive branches of the Toda and real potentials coincide. Thus the effects of anharmonicity and nonconvexity can easily be distinguished. The broken curve gives the harmonic approximation, which is common to both potentials. The dashed vertical line through the inflection point ri separates the regions of distortion and undistortion, also shown at the bottom.

15

proximation (the dashed line in Fig. 13), which for the generalized Morse potential (eqn. (35)) reads

V(r) = &/NV& - .>z - V, (37)

The product E=~vV~ gives the force constant, which is a measure of the bonding between the overgrowth atoms. Obviously, the harmonic approximation can be used for small deviations from the equilibrium atom separation, i.e., for small strains r-r,. This is equivalent to small misfits in interface problems.

Let us analyze more closely the above pair-wise potentials. Figure 14(a) demonstrates the variation with atom spacing of the first derivative of the pairwise potentials or the force exerted on an atom by its neighbour. As seen, the force goes linearly to infinity

Fig. 14. First (a) and second (b) derivatives of the generalized Morse potential (referred to as a ‘real’ potential), the Toda potential and the harmonic approximation. The first derivative gives the interatomic force, whereas the second derivative determines the sign of the curvature.

in the harmonic case. This means that increasing the atom spacing leads to an increase in the force, which tends to keep the atoms together. Hence a harmonic chain can never be broken. The Toda force, F(r) = &,{l - exp[ - p(r - r,,)]}, however, goes to a constant value N, at large atom separations. This means that by applying a force greater than the maximum one, the corresponding bond can break up and both atoms can be separated from each other. The same is valid for potentials (33)-(35) (henceforth referred to as real potentials), where the force displays a maximum. The latter can be considered as the theoretical tensile strength of the material.

Figure 14(b) demonstrates the variation of the second derivative of the pairwise potentials, which in fact determines the sign of the curvature. In the harmonic case the second derivative is constant and positive; in the Toda case it is a decreasing function of the atom separation and goes asymptotically to zero but always remains positive. Only in the case of the real potentials does the curvature change sign from positive to negative at the inflection point r=ri. In other words, the real potentials become nonconvex at r>ri. As shown by Haas [31, 321 the nonconvexity of the real potential results in a distortion of the chemical bonds in an expanded chain (or epilayers): long, weak and short, strong bonds alternate (Fig. 13). The driving force of such a distortion is the energy difference between the distorted and undistorted structures. It is easy to show that the mean energy of a distorted (dimerized) structure (Fig. 13) is [V(r + U) + V(r- u)]/2 < V(r) for a curve with negative curvature, [ V(r + u) + V(r - u)]/2 = V(r) for a straight line (zero curvature) and [V(r + u) + V(r - u)]l 2> V(r) for a curve with positive curvature. It follows that when applying the harmonic potential or the real potential at misfits smaller than that corresponding to the inflection point ri the ground state will be the undistorted structure. The distorted structure will be the ground state in epilayers expanded beyond the inflection misfit when a real potential is adopted.

3.2. Interfacial interactions A single atom moving on a single crystal surface

should feel a two-dimensional periodic potential relief [17]. Assuming a corrugation in one direction only, the potential can be written in the form

W(X)=G+-cos 2G]

which was initially introduced by Frenkel and Kontorova

]771. The amplitude W is related to the substrate-deposit

bond strength by

W= B(pd (39)

16

where (pd is the desorption energy of an overlayer atom from the substrate surface and 9 is a constant of proportionality varying from l/30 for long-range van der Waals forces to approximately l/3 for short-range covalent bonds [13]. In fact, W is the activation energy for surface diffusion and p gives the relation between the activation energies for surface diffusion and desorption.

3.3. Effect of anharmonicity in epitaxial inte$aces We make use of the Toda potential (36) and the

substrate periodic potential (38). Then eqn. (28) gives the potential energy of an anharmonic (Toda) chain consisting of & atoms [27, 281.

The disparity in the structural properties of the overgrowth with respect to the sign of the misfit is clearly demonstrated in Fig. 15 [28]. It shows the split of the limits of stability fs and metastability fms with respect to the misfit sign. The increasing degree of anharmonicity p results in a reduction in the values of fs and fms for compressed chains (b >a) and an increase in the absolute values offs and fms for expanded chains (b <a). The respective values for the harmonic model are given by the dashed lines. Thus the harmonic limit of stability, fl= +8.6% [18], splits into +6.7% and - 12.2%, whereas the limit of metastability, f& = f 13.6%, splits into + 10.2% and - 23.2% at some average degree of anharmonicity p= 6. Therefore a pseudomorphous overlayer can be in a state of stable (below fJ or metastable (below fmfms but above fJ equilibrium up to quite different limits at positive and negative incompatibility with the substrate.

E 3 26

0 2 4 6 P

ANHARMONICITY

Fig. 15. Variation of the limits of stability f, and metastability f,,,# with increasing degree of anharmonieity p for long-enough Toda chains and different signs of the misfit. The reference values of the harmonic model of Frank and van der Merwe [18],

fl andf!h are shown by the dashed lines.

A very important conclusion that follows from the split of the critical misfits with respect to the misfit sign is connected with the critical thickness for pseudomorphous growth. The latter is qualitatively pro- portional to the square of the stability limit fs (t,=bCf,/ f)’ [78]). It should be expected that the critical thickness for pseudomorphous growth will be 3-4 times greater when the natural misfit is negative rather than positive if all other parameters remain unchanged. This prediction of the model seems particularly important for the epitaxial growth of semiconductor films and strained- layer superlattices, where the dangling bonds associated with the MDs [79] have a deleterious effect on the performance of the devices based on the corresponding heterojunctions [80]. Thus MBE-grown In,Ga,_Js on (100) InP shows asymmetric behaviour of the critical thickness for pseudomorphous growth with the sign of the misfit [23, 241, the thickness of expanded epilayers being always greater. The same is observed in LPE- grown In,Ga, _&,,P, _y on (100) InP [25].

It also follows from the above that when the lattice misfit has different signs in different crystallographic directions the film can be partially pseudomorphous. The partial pseudomorphism is defined as a state in which the interface is resolved in an array of MDs lines in one direction only instead of a cross-grid of MDs. An excellent example is the structure of the interface formed between MoSi, and WSi, on (111) and (100) faces of Si, respectively [81, 821. As seen in Table 1, partial pseudomorphism (denoted in the table by PP) is observed not only when the lattice misfit is positive in one direction and has a negligible value in the other, but also when the misfit has a positive value in one direction and a negative one in the other. In the case denoted by E for tetragonal t-WSi, the misfits in both directions have the same absolute value but have opposite signs. In the E case of the tetragonal MoSi, the absolute value of the negative misfit is even greater than that of the positive misfit and the epitaxial interface is still resolved in an array of parallel MD lines but not in a cross-grid of MDs. When the misfit in both directions is negative the films are completely pseudomorphous (CP) with the substrate. Cross-grids of MDs (denoted by MD) are observed only in cases when the misfit in both directions is positive and not negligible.

Using the embedded atom method [83] Dodson concluded that the critical thickness for pseudomorphous growth in bimetallic systems should be greater for compressed than for expanded overlayers [84]. He found, for example, that the critical thickness for Au/Pt(lOO) cf= + 4.08%) is two monolayers, whereas for Pt/Au( 100) cf= -3.92%) it is one monolayer. However, Murthy and Rice [85], using the same method in the case of Cu/Ni(lOO) and Ni/Cu(lOO) systems, concluded that

17

TABLE 1. Structure of the epitaxial interfaces between MoSi, and WSi2 and Si(OO1) and Si(lll)a

Deposit Epitaxial orientation Direction Misfit Structure

a b %a %b

t-MoSi,

A B C D E

h-MoSiZ

A B C

t- WSi, A B C D E

h- WSiz

A B C

[lie] 2.34

[liO] 2.21

[lie] 2.34

[loi] 2.34

[loi] 2.21

ProI 11101 1r101 [lie] [iio]

- 1.69 - 1.69

0.10 2.21

- 2.68

PP PP PP MD PP

P101 pJ1 WI

[iio] 4.04

10101 - 2.89

[ioi] 4.04

0.13 PP - 1.84 CP

4.04 MD

11101 11101 11101 [lie] [lie]

[lie] 2.47

[iio] 2.47

[iio] 2.47

[loi] 2.47

[loi] 2.47

- 1.43 -1.43

0.37 2.47

- 2.46

PP PP PP MD PP

[iio]

10101 [loi]

PlOl p1 PlOl

4.04

- 3.01

4.04

0.13 PP - 1.84 CP

4.04 MD

‘t and h refer to tetragonal and hexagonal, respectively. PP: partial pseudomorphism (array of MD lines); CP: complete pseudomorphism; MD: cross-grid of MDs [81, 821.

expanded overlayers should grow pseudomorphous with the substrate to a greater thickness than compressed ones.

Figure 16 shows the characteristic split of the misfit dependence of the mean dislocation density fd with respect to the misfit sign [28]. As seen, f; is always smaller than fi, although the difference gradually decreases as the natural misfit increases. It can also be seen that the harmonic approximation is much closer

to the positive misfit curve. What is more important, however, is that the curves, although shifted from the harmonic one, preserve their continuous character. In other words, the transition from the pseudomorphous cfd =0) to the completely dislocated cfd =f) state is continuous and there is a misfit interval in which both HS and MDs coexist.

The misfit dependence of the ground state energy per atom is shown in Fig. 17 [28] for both positive (dashed line) and negative (solid line) misfits. The curves consist of a series of curvilinear segments, as in the harmonic case. The segments correspond to different numbers of MDs, increasing from zero by one. It is seen that in the case of a compressed chain and particularly at small misfits, the energy is considerably higher. At larger misfits the energy curves become closer and eventually merge. At low misfits, both positive and negative, the strain energy is dominant. At positive misfit the steeper repulsive branch of the interatomic potential is mainly involved and the energy is accordingly higher than in the case of negative misfit, where the strain energy is determined by the weaker, attractive part of the interaction. At larger misfits, both positive and negative, the interfacial bonding in eqn. (28) pre- dominates and the energy difference gradually vanishes. It is worth noting that the harmonic curve (not shown in the figure) is again much closer to the positive misfit curve. The above result is in agreement with the finding of Murthy and Rice [85] that the interface energy of

MlSFll

Fig. 16. Variation of the mean dislocation density fd with misfit. The dashed and solid lines refer to positive and negative misfits, respectively. The dotted line gives the harmonic reference in the continuous limit [18]. Both cases of positive and negative misfits are shown in one quadrant for easier comparison. In fact, the figure shows the dependence of the average atomic spacing 6 on the natural spacing b. The straight line refers to 6 =b.

Fig. 17. Variation of the ground state energy per atom of the Toda chain with misfit. The dashed and solid lines refer to positive and negative misfits, respectively. The number of MDs in the ground state is shown in the figure at each curvilinear segment.

the low, positive misfit couple Cu/Ni(lOO) cf= + 2.56%) is considerably greater than that of the negative misfit couple Ni/Cu(lOO) cf= - 2.49%).

It can be concluded that negative misfit appears to be more favourable than positive misfit for the epitaxial growth of thin films. If several epitaxial orientations are possible for a particular overgrowth material on one and the same substrate plane, the orientation connected with negative misfit should be favoured. An example for that is the orientation of Ag on (OOi) GaAs [86]. At temperatures lower than 200 “C the epitaxial orientation is (llO)[lll],,~](OOi)[ilO],,,, with fx = 2.23% andf, = - 27.7%. However, four atomic spacings of Ag nearly coincide with three atomic spacings of GaAs, and the lattice misfit is expressed as the relative difference of the multiple atomic spacings: f, = (4b - 3a)/3a = - 3.62%. At temperatures higher than 200 “C the overgrowth is in parallel orientation - (OOl)[OlO],,]](OOi)[O1O],,,, - and the lattice misfit in both orthogonal directions is negative: fx =f, = - 3.62%.

3.4. Influence of nonconvexity on epitaxial inter$aces The effect of anharmonicity can be more or less

intuitively predicted from the asymmetry of the interatomic potentials. This is not, however, the case with the more realistic potentials (33)-(35), where the nonconvex character leads to the existence of a maximal force between the atoms at the inflection point and to distortion of the chemical bonds when the latter are

stretched out beyond the inflection point. That is why we will consider this case in more detail.

3.4.1. Model In order to study the effect of the nonconvexity of

the real potentials, we make use in eqn. (28) of the generalized Morse potential (eqn. (35)). The latter has an inflection point

(40)

beyond which the second derivative d2Vldr2, which determines the sign of the curvature, becomes negative and has a minimum at

The value of the minimum

(41)

determines the maximum driving force for distortion to occur.

The equilibrium condition aE/&X,, =0 results in the following system of recurrent equations, giving the equilibrium displacements of the atoms:

expE-cca(5~-5,-nl-exp[-~(5~-~~-f)l = -A sin 2rr&,

exp[-p(f+,- k-f)1 -exp[- ~(5n+I-k-f)l

-exp[-~(5,-Sn-1-nl+exp[-va(5,-5,-1-f)l = -A sin 2~&, (43)

exp[-lla(Z,-,-5N-2-f)l-exp[-~(~-,-5,-2-f)l = A sin 2rrtN_,

where

(44)

and &,=X,/u are the relative displacements. It is easy to show that at small strains eqn. (43) turns into the static sine-Gordon equation in the continuum limit.

3.4.2. Distortion of the chemical bonds We consider first an infinite chain. The undistorted

state is one in which all atoms are equally spaced at a distance equal to the substrate potential period a (Fig. 18(a)). A distorted chain can be dimerized so that short and long bonds alternate (Fig. 18(b)). As mentioned above, this phenomenon is due to the fact that the average energy of one long and one short bond is smaller than the energy of a bond of intermediate

Fig. 18. Distortion patterns: (a) undistorted state; (b) dimerized state; (c) trimerized state; (d) tetrarnerized state; (e) pentamerized state.

aL A

UNDISTORTEO

0 03 06 MlSFll

Fig. 19. Phase diagram of existence and stability of distorted and undistorted states. Curve A outlines the area of existence and stability of the dimerized state. Curve B outlines the region of existence of trimers. Curve C divides the regions of stability of dimerized and trimerized states.

length [31, 321. In a dimerized chain the displacements of the consecutive atoms are equal in absolute value and opposite in sign: &, + 1 = - [,, = &, _ 1.

Obviously, a strong substrate-deposit interaction (WZ+ V,) favours the undistorted structure. A distorted structure will be tolerated when the ratio WlV, is small enough. Thus, applying the condition &,+1 = - [,, to eqn. (43) for W/V, in the limit m--+0, one obtains

This dependence of critical substrate modulation WC/ V, on the critical misfit fc outlines an area in which the dimerization is energetically favoured. It is plotted in Fig. 19 (curve A). As seen, it starts at fc=fi, given

by

r,-ri W-h9 fi=-=__ a ah- 4

and displays a maximum at fc=fm, where

(46)

To-r, fm,-...---=- a

2 w-w =2f, ah.-4 ’

19

(47)

The maximum value

corresponds to the maximum driving force (eqn. (42)) for distortion to occur. Clearly, dimerization cannot take place when W> Wm.

We consider further the formation of trimers or alternation of two equally short bonds and one long bond (Fig. 18(c)). Within the trimer &,+ 1 = - [n_1 and 5, = 0. The curve that outlines the area of existence of trimers lies under that of dimers (Fig. 19, curve B), its maximum value at f = fm being given by

m _ 3p2a2 v 2p’(p- “) W

0 v. 27r2 j_L (49)

Curve C in Fig. 19 separates the regions of stability of dimers and trimers. Below it the dimers still exist but not as a ground state.

In the same way we consider tetramers, pentamers, etc. (Fig. 18(d) and (e)). By repeating the same pro- cedure we find that the regions of existence of polymers with degree of polymerization higher than 2 are included in that of dimers. It follows that curve A separates the regions of stability of distorted and undistorted states. It can be concluded that at W> W,,, no distortion of infinitely long chains take place, irrespective of the value of the natural misfit. Besides, the higher the degree of polymerization, the smaller the value of W at which the corresponding polymers are energetically favoured. In the limit W-+0, the degree of the energetically favoured polymers goes to infinity, which in practice means the disappearance of the distortion. This is equivalent to alternation of an infinite number of short bonds and one long bond, which in fact means an undistorted structure.

As seen in Fig. 19, the undistorted state (5, =0) is the most favourable state at low absolute values of the negative misfit and strong interfacial bonding. When the misfit is increased above a certain critical value fc, determined by eqn. (45), a transition to a distorted dimerized state (5, ~0) takes place. One can consider the absolute value of the displacement j&j as an order parameter characterizing the dimerization of the bonds. It was found that close to line A in Fig. 19, the order parameter ]&I behaves like ]&,I -(f-f$” [30]. The second derivative of the energy with respect to f is discontinuous at f= f,. Since T= 0, the potential energy replaces the free energy and the distortion is a second- order transition. The order parameter ]&J has the same critical behaviour with respect to the interfacial bonding,

20

IS~l-(K-W’“, and the second derivative of the energy with respect to W is again discontinuous at W= WC, i.e., the distortion is a second-order transition with respect to both the misfit and the interfacial bonding. Note, however, that this is not a ‘true’ phase transition in the thermodynamic sense, as T=O and hence there are no thermal fluctuations in the system to induce a phase transition.

It is worth noting that a two-dimensional distortion of the chemical bonds (clustering of 4 or 8 atoms) has been theoretically found with the help of the embedded atom method in a Ni monolayer grown on Ag(lOO) by Bolding and Carter [87]. The absolute value of the negative misfit is very large: f= - 13.9%. The growth of the second monolayer causes a relaxation of the distortion of the first-layer bonds, i.e., the atoms of the first monolayer tend to occupy the bottoms of the potential troughs of the silver substrate. However, the bonds in the second monolayer become distorted but their distortion is weaker. After the deposition of four monolayers the distortion of the bonds between the atoms of the first monolayer practically vanishes. Thus the Ni atoms closest to the substrate are under the largest uniform expansive strain and the strain dimin- ishes away from the contact plane. Rajan [88] interpreted the distortion of the chemical bonds in terms of the ‘coherency or transformation dislocations’ which are required to achieve coherency of the lattice planes in contact at the epitaxial interface (see also ref. 89).

We consider now a chain of finite length. It will be distorted if appropriate values of W/V, andfare selected (Fig. 19). If this is not the case, the middle part of the chain will not be distorted, but it turns out that the bonds near the ends of the chain are always distorted

as long as VI > Ifit, irrespective of the value of W/V, The latter is evidently due to the asymmetry of the atomic interactions near the free ends. As will be shown below, this edge effect leads to significant results concerning the metastability limit of the pseudomorphous state and the activation energy for introduction of dislocations at the free ends.

3.4.3. Ekistence of solutions Inspection of eqn. (43) concerning the existence of

solutions is difficult in the general case, as the equations are not solvable with respect to the highest variable therein. That is why we consider the simplest case of a Morse chain @=2-y, V= r), bearing in mind that examination of other cases (e.g., p = 3, u= l), although more complicated, leads to the same conclusion.

The set of difference equations governing the behaviour of a Morse chain can be written in terms of strains E,, = &-(~-I-f instead of displacements & in the form [29]

El”-fi- ‘In If ya

[ JIIW]

c&+1= -fi- $

Xln l+ l-4 e-rrrti-e-2w*+ [J (

5TW - - sin 2rr&

2ravO )I (50)

l N_r= -fi- $ln [ d-1 l+ l+

where

r,-ri In 2 fi=_=--

a 7a

is the misfit that corresponds exactly to the inflection point of the Morse potential.

One may look for a solution of equations (50) (and hence of eqns. (43)) provided the logarithmic terms therein are well-defined analytical functions, i.e., when their arguments are non-negative. This condition is fulfilled when the discriminants D under the square roots are positive for positive signs before the square roots, or positive but smaller than unity for negative signs before the roots. When the sign before the root is positive for D > 0, the corresponding strain .E,,+~ is always smaller than -fi. In the other case, of negative sign for 0 <D < 1, the strain E,+ 1 will be greater than -fi. This emphasizes the fundamental role that the inflection of the real potential plays. The latter becomes clearer when distortion of the chains at negative misfits takes place. When the strains of the short and long bonds are smaller and greater than -fi, respectively, positive and negative signs alternate in the consecutive equations of eqns. (50).

The condition for the existence of solutions D> 0 leads to the inequalities

f YV, > 2yV0(ePpo’ -e-2yo*) + F sin 2~& (51)

f yVO > - kw sin 2~&_ 1

which means that in order to have solutions of the system of eqns. (50), the resultant force exerted on the nth atom by the (n- 1)st atom (the first term on the right-hand side of eqn. (51)) and by the substrate (the sine-containing term on the right-hand side of eqn. (51)) must be smaller than the theoretical tensile

21

strength, a,,, = rV0/2, of the Morse potential. If this is not the case, the corresponding bond will break up and the chain will lose its integrity.

This is just what happens to the most expanded bonds in the cores of the MDs at positive misfit. The MD core bonds are expanded, and when their strains cc become equal to -fi, or in other words when the force exerted on the bonds exceeds a,,,, the chain breaks up. Figure 20 represents the strain ec of the bonds in the cores of the dislocations as a function of the natural misfit. At some critical misfit the core strain cc reaches -fi and the chain breaks up just in the dislocation core. This does not mean that bonds that are stretched out more than -fi cannot exist. As will be shown below, in the case of chain distortion, bonds dilated much more than -fi can exist without rupture. The explanation can easily be seen by looking at Fig. 21(a). In

Fig. 20. Variation of the strain in the cores of the MDs vs.

positive misfit at variousvalues of the relative substrate modulation

W/V, given by the number on each curve.

(b) -a-

Fig. 21. Configurations of misfit dislocations at (a) positive misfit

(light wall) and (b) negative misfit (heavy wall).

the case of positive misfit the MD represents an empty trough and the atoms on both sides of the core bond are located in such a way that the force exerted by the substrate is destructive. It is just the opposite in the case of distorted chains at negative misfits greater in absolute value than -fi. The force exerted by the substrate on the atoms on both sides of the more expanded bonds is not destructive but tends to keep them together (see Fig. 18).

3.4.4. Solutions As discussed above, the real potentials are divided

into two parts: a convex part when r < Ti and a nonconvex part when r>ri. The convex part contains the region of positive misfits and the region of comparatively small (in absolute value) negative misfits. When the particles of the chain experience the convex interactions (~90 or Ifl <b]), the solutions of system (43) are in fact the same as in the harmonic case. Except for the trivial solution 5, = 0 (complete accommodation of the lattice misfit by HS, i.e., pseudomorphism), system (43) has solutions satisfying the condition &._ 1 - &, = *IV,, where ND is an integer equal to the number of MDs (single solitons) in the chain. The positive sign holds for positive misfits when the MDs represent empty potential troughs [Fig. 21(a)] (positive MDs or light walls in the theory of CI transitions [65]), whereas the negative sign corresponds to negative misfit when the MDs represent two atoms in one trough, or three atoms in two troughs [Fig. 21(b)] (negative MDs or heavy walls).

The situation changes dramatically when the chain particles sample the nonconvex part of the potential. Instead of being equidistant, the atoms are clustered in groups of two, three or more atoms. The clusters are connected by bonds that are expanded more than the inflection distance ri. The conditions for existence (eqn. (51)) are fulfilled and the distorted configurations are stable. In fact, the distortion leads to periodic variation of the strain, with a period equal to the degree of polymerization (Fig. 22). Note that the strains do not change sign. The chain remains pseudomorphous with the substrate, as the condition for coherency h=a (fd = 0) is not violated. If the chain contains MDs (fd # 0) the bonds between the dislocations are also distorted (Fig. 23(a)). Th e introduction of MDs into the chain leads to periodic variation of the strain, with a period equal to the MD spacing. In this case the strains change sign, so that positive and negative strains alternate (Fig. 23(b)). Thus two kinds of periodicities coexist, one due to the MDs and the other to the distortion of bonds.

Of particular interest are the solutions when the bonds in the cores of the MDs are distorted. At large absolute values of the misfit and weak interfacial bonding, coupled negative-positive MD (soliton-antisoliton

22

:::: i/ 40 50 60

(a) SPRING NUMBER

0.6 2

0.6

z

3 0.4

k

0.2

0.0 0

(b) SPRING NUMBER

Fig. 22. Variation of the strains of the chemical bonds in a coherent distorted chain: (a) dimers cf=fm = - 17.33%, W/V,= 3.0, p = 8, v= 4, a = 2); (b) a sequence of mixed hexamers, heptamers, etc. cf= -9.5%, W/V,=O.2, ~=8, v=4, a=2).

0.5

E i-2 -0.0

g -0.5

$0

E-1.5

E-2.0 d -2.5

-3.0 0 10 20 30 40 50 60

(a) ATOM NUMBER

0.3

(4

1.5

-0.1

-0.2 0

('-') SPRING NUMBER

-1.0 0

(b)

10 20 30 40 50 60 70 60 90 SPRING NUMBER

Fig. 23. Variation of (a) the atomic displacements 5, with atom Fig. 24. Variation of (a) the atomic displacements 2, with atom number and (b) the bond strains E, with spring number in a number and (b) the bond strains 6. with spring number in a distorted (tetramerized) chain resolved in a sequence of single distorted chain resolved in a sequence of multisolitons. Note the misfit dislocations cf= -lO.O%, W&=1.0, p=8, v=4, a=2). similarity between the multisolitons and the bond distortion.

or multisoliton) configurations are observed [30]. The atom displacements and the bond strains of such configurations are shown in Fig. 24(a) and (b), respectively. The larger the misfit and the smaller the substrate modulation W, the greater the number of single solitons and antisolitons involved. As seen in Fig. 24(a) the antisoliton consists, in fact, of only one empty potential trough. The force exerted by the substrate is not destructive (- 1.5 < &, < - 1.0 and 0 < .$n+I <OS); it keeps the atoms together. Although the condition for the existence of solutions (eqn. (51)) is not violated, the force exerted by the substrate exceeds but is very close to the force exerted by the neighbouring atoms. As a result the expanded bond in the core of the antisoliton is on the verge of breaking. One could consider such antisolitons as nuclei of the cracks in expanded epitaxial overlayers. Very often coupled soliton-antisoliton- soliton (SAS) configurations are observed, as shown in the right-hand side of Fig. 24(a). They can be separated from soliton-antisoliton (SA) configurations (left-hand side of Fig. 24(a)) with distorted bond clusters, as seen in Fig. 24(a), but not always. SA and SAS configurations can merge to give rise to SASAS (or even SASASAS) configurations at large absolute values of the misfit. With increasing substrate modulation W the bond distortion becomes less pronounced (the force exerted by the substrate considerably exceeds the destructive force exerted by the neighbouring atoms) and the multisoliton

0.4

2 3 0.0 z

fo.,

S-O.6

2

s-1.2

ATOM NUMBER

solutions disappear. Figure 24(b) shows the consecutive strains in a chain with distorted bonds and SA solutions. The similarity in bond distortion between the solitons and the SA configurations is evident.

3.4.5. Energy The behaviour of the energy versus misfit of a real

chain differs qualitatively in comparison with that in the harmonic case, particularly at positive misfits. At small values of W (W/V, <OS) the positive misfit dependence of the energy is similar to the harmonic one and to that of the anharmonic Toda chain (Fig. 17). At larger values of W, however, the E(f) dependence consists of curvilinear segments that do not intersect each other (Fig. 25(a)) owing to the rupture of the core bonds at small values of the misfit. This tendency becomes more pronounced with increasing W (Fig. 25(b)), and in the case of shorter chains the effect of core bond rupture is so strong that the segments do not overlap and gaps appear between them in which no solutions of system (43) exist. This is seen more

0 0.05 0.10

(4 MISFIT

@I ’ 0 -005 -015 -0 25

Fig. 25. Variation with positive misfit of the energy per atom in

units of W (a) W/Va=O.S; (b) W/I/,=4 (N=80, ~=4, v=3). Fig. 27. Variation with negative misfit of the energy per atom:

The numbers on the curvilinear segments denote the number Curve A, chain without a dislocation; B, chain containing one

of dislocations. dislocation (W/V, = 1, N = 100, p= 4, v= 3). Note that there are

no singular points at the inflection misfit f;.

clearly in Fig. 26, where the dependence of the energy on chain length is shown at a constant value of the positive misfit. Gaps without any solution exist for short chains, but disappear for longer chains. The energy shows a sawtooth behaviour, and the introduction of each new dislocation is connected with an abrupt energy drop, which is uncharacteristic of the harmonic model.

The case of negative misfit is quite different. The core bond strain is compressive and the force exerted by the substrate is not destructive (Fig. 21(b)). It follows that expanded epilayers cannot break up in the cores of the dislocations at negative misfits. On the other hand, at misfits larger in absolute value than fi the bonds are distorted in between the dislocations (Fig. 23) and rupture there is again excluded. The appearance of cracks in expanded epilayers could be expected at large values of the misfit as a result of the rupture of bonds in the cores of the antisolitons.

Figure 27 shows the energies of chains without a dislocation and containing one dislocation. The chains are distorted at Ifl> FJ. Both curves intersect with each other at the limit of stability,f;. As seen, the energies are very close, particularly at misfits greater in absolute value than fi. Obviously, the contribution of the dislocation energy is small compared with the contribution

06 L

1

0 10 20 30 40 50

NUMBER OF ATOMS

Fig. 26. Dependence of the energy per atom in units of W on

the chain length. The number of dislocations in the ground state

is denoted by the number on each segment (W/I/,=4, f=0.2, /.&=4, v=3).

24

of the chain distortion. Note that no singularity is observed at the inflection misfit fi, above which bond distortion takes place. This is obviously a consequence of the second order of the distortion transition discussed above.

However, at negative misfits there are two peculiarities that are uncharacteristic of the harmonic case. First, at strong enough bonding across the interface (IV/ V0>W,lV0=2), th e energies of the chains without a MD and containing one MD no longer intersect. The energy of the incommensurate state asymptotically approaches the energy of the commensurate state, being always greater than the latter. It follows that the limit of stability disappears at strong interfacial bonding, and epilayers that are thin enough to fulfill the requirements of the model will always be pseudomorphous with the substrate, irrespective of the absolute value of the natural misfit.

The second consequence of the application of real potential to the 1D model of Frank and van der Merwe, which is absent in the original model, is that solutions of coherent configurations, although not in a ground state, exist at any value of the negative misfit provided the potential troughs are deep enough (W/V,> W,,l V,= 0.25, Ifl> bj). Th is is due to the distortion of the free ends of the chains even under conditions (W, f) in which the ground state is the undistorted state (Fig. 16). Owing to the chain end distortion, the end atoms do not climb the slopes of the potential troughs with increasing misfit, which excludes the possibility of spontaneous introduction of dislocations at the free ends. It follows that the metastability limit fis of the pseudomorphous state disappears at W> W,,,,.

The above leads in turn to the obvious conclusion that at I> KI the activation energy for introduction of a dislocation at the free ends is greater than that in compressed chains. Owing to the chain distortion, the end atoms of the chain are always near the bottoms of the potential troughs. Hence the introduction of a new dislocation requires overcoming a much higher energy barrier than in the case of positive misfit, particularly at stronger bonding across the interface. It follows that expanded overlayers can exist in a metastable state without dislocations at higher temperatures compared with compressed epitaxial films.

The potential anharmonicity and the bond distortion result in a split of the energy of interaction between the MDs (Fig. 28). The latter is considerably larger for a positive misfit than for a negative one (Fig. 28(a)). The data from Fig. 28(a) are plotted on a semiloga- rithmic scale in Fig. 28(b). As seen, for a positive misfit, both in the real case and in the harmonic limit, the plot deviates slightly from linearity at high MD densities (small MD spacing). In expanded chains the MD interaction energy depends exponentially on the MD

MD NUMBER

I I

10.2 200 300

(b) MD SPACING I~nol

Fig. 28. The behaviour of the MD pair interaction energy: (a) dependence of the energy on the number of the MD for positive and negative values of the natural misfit; (b) logarithmic plot of the energy of MD interaction against the MD spacing (reciprocal of the MD density). Curves 2 in both figures represent the harmonic limit (N= 60, f= f 7%, W/V, = 0.5, p = 4, V= 3).

density, even at high values of the latter, which is an indication of weak interaction [65]. This is obviously due to the fact that the MD interaction is realized through the weaker attractive branch of the interatomic potential in expanded chains and through the steeper repulsive branch in compressed ones. The bond distortion in between the MDs contributes additionally to the weak interaction between them in expanded chains.

As shown by Markov and Trayanov [30], the data from the numerical solutions fit the semiempirical expression

E=i?YOddewffi ( ) l- f +8(O) s

(52)

This expression describes all the data surprisingly well. Moreover, the harmonic limit [ 181, for which& = 03, is also formally included. Then, by analogy with the harmonic model [18], we can write the following expression for the energy of a single MD:

gd=gpe-f/f' (53)

where 8; is the energy of a single dislocation at f= 0. Since fi<O, it follows that the MD energy 8, is a decreasing function of the negative misfit and an increasing function of the positive misfit.

The zero energy of a single dislocation, Sy, is shown in Fig. 29 as a function of the quantity (/_~vV,w)“~ (I/,=const.). The straight line (curve 1) represents the harmonic reference. Curve 3 gives the energy of a negative dislocation (two atoms in a trough, Fig. 21(b)), whereas curve 2 gives the energy of a positive dislocation (an empty trough, Fig. 21(a)). The difference of the energies clearly reflects the anharmonicity of the real potential. In expanded chains (negative misfit), the atoms in the MD core interact through the steeper

25

Fig. 29. Dependence of the zero energy cf=O) of the static solitons on WIR. The straight line (curve 1) presents the harmonic limit of Frank and van der Merwe [18]. Curve 2 gives the energy of a positive MD (an empty trough or a light wall, Fig. 21(a)), and curve 3 shows the energy of a negative MD (two atoms in a trough or a heavy wall, Fig. 21(b)).

repulsive branch of the potential and the zero energy 8; is greater than that in compressed chains, where the weaker attractive branch operates.

3.4.6. Limits of stability The form of the E(f) dependence shown in Fig. 25,

which is due to the rupture of the dislocation core bonds, leads to a new definition of the limit of stability, fc, of the pseudomorphous state in compressed films. It is now determined by the condition of existence of a dislocation with core bond stress smaller than the theoretical tensile strength and coincides with the critical misfit for rupture of the most expanded core bonds.

As shown in Fig. 15, the limits of stability and metastability split with respect to the sign of the misfit when anharmonicity is ‘switched on’. Figure 30(a) illustrates the split of the stability limit fS as a function of the substrate modulation W when a nonconvexity is added to the anharmonicity of the interatomic potential. The harmonic case is given by the straight line, denoted byf,h. The corresponding curves for a Toda chain (a= 2, p= 6) are also included for comparison. As seen, the positive stability limit, fc, lies nearer to the reference harmonic curve than the anharmonic Toda curve. This is not surprising if we keep in mind that in the real model at positive misfit, the stability limit of the pseudomorphous state is determined not by the equality of the energies of states with zero and one dislocation but by the rupture of the MD core bonds. The latter takes place at larger values of the misfit. This is the reason f: lies nearer to the harmonic reference than the Toda curve. On the other hand, the negative stability limit, fS-, is shifted to greater absolute values than in the Toda case, which is easily understandable considering the shape of the corresponding attractive branches of both potentials (Fig. 13). What is more important is that fSP disappears after some critical substrate-deposit bond strength W= W,=2V, (note the x

0 -- 15 20

(4 (W/VJZ

,Ot f,Jlodol

#’ I’

,’ ,’

#’ ,’

h fm.

Fig. 30. Variation of (a) the limits of stability, f,, and (b) the limits of metastability,f,,, with (W/Vo)‘R (~=4, ~=3). The straight lines, denoted byf: and&, show the harmonic limits. The limits of stability and metastability of the anharmonic Toda model (a = 2, /3 = 6) are also included as broken lines for comparison. The negative stability limit f; terminates at W/V,=2, whereas the negative metastability limit f,& terminates at W/V, = 0.25 (see the corresponding x signs).

sign at the corresponding curve). Beyond this value the pseudomorphous state is always the ground state.

Contrariwise, the positive metastability limits,f,‘,, for the real and Toda potentials (Fig. 30(b)) overlap, which reflects the coincidence of the respective repulsive branches, as seen in Fig. 13. However, f,& disappears beyond some critical value of the potential amplitude W,,,,/V,=O.25 (note the X sign) governed by the con- ditionf;,( W,,) =fi. As mentioned above, the chain ends are distorted and spontaneous generation of dislocations at the free ends never takes place, whereas the Toda metastability limit still exists.

3.4.7. Mean dtilocation density The mean MD density fd in the ground state of a

real chain is given in Fig. 31 as a function of the natural misfit f [30]. Both curves, for positive and negative

26

z s z 015 0

i=

iz JI 0 z 005

d ?I

0 005 015 025

MISFIT

Fig. 31. Plot of the mean MD density in a real chain against the natural misfit for both positive (left) and negative (right) misfits. The harmonic continuous approximation of Frank and van der Merwe [lg] is presented (smooth curve) for comparison. The two curves are plotted in the same quadrant for easier comparison (~=4, v=3, W/VO=l, N=60).

values of f, are presented in the same quadrant for easier comparison. The smooth curve represents the continuum limit of the harmonic model [18]. The step- wise behaviour is due to the finite size of the chain (iV=60). The splitting of both curves around the harmonic reference is due to the anharmonicity of the real potential. The positive misfit curve is considerably nearer to the harmonic limit than the respective curve for a Toda chain, which is due to the limited interval of existence of the dislocated state as a result of the rupture of the most expanded bonds in the cores of the MDs in compressed chains. What is more interesting is that the CI transition is continuous in compressed chains but abrupt in expanded ones, going in a single jump from zero to the maximum density of the MDs.

The continuous behaviour of the CI transition is due to the energy of the dislocation interaction [18, 651. As shown in Fig. 28, the energy of MD interaction is much smaller in expanded chains than in compressed ones. This explains the abrupt behaviour of the CI transition in expanded chains.

3.4.8. Comparison of the models It is thus evident that in addition to its anharmonicity,

a fundamental characteristic that distinguishes the nonconvex interatomic potential from the harmonic approximation is not the finite energy of dissociation of two neighbouring atoms but its inflection. The latter leads to two effects: (1) the existence of maximal force between the atoms in the chain, which can be considered as the theoretical tensile strength of the material, and (2) the distortion of the chemical bonds in expanded epilayers.

The anharmonic Toda model has in fact the same solutions as the harmonic approximation. New solutions (distortion of the bonds and multisolitons) appear when the nonconvexity is ‘switched on’. As a result of the

distortion of the chemical bonds, the concept of coherency or pseudomorphism should be reconsidered [88, 891. One can no longer say that the MDs divide the interface plane into regions of perfect match. The condition of coherency, b=a (fd= 0), is still fulfilled, but a periodic variation of the strain appears. The latter is similar to the periodic variation of the strain associated with the MDs (Fig. 24(b)). The appearance of static multisoliton solutions (not to be confused with the ‘breathers’ in the classical sine-Gordon chain [90]) is a consequence of the distortion of the bonds in the cores of the MDs. The usual dislocation (heavy wall) splits into two dislocations with a strongly expanded bond (antisoliton or a light wall) between them. The strain of the antisoliton core bond is determined by eqn. (50), with a minus sign before the square root.

The anharmonicity of the atomic interactions leads to the split of all the properties (the ground state energy, the limits of stability and metastability, the mean dislocation energy, etc.) of the classical model with respect to the sign of the misfit. However, their behaviour remains ultimately the same. Thus the ground state energy of a finite chain consists again of curvilinear segments that intersect each other. As a consequence, the limits of stability are defined in the same way as in the harmonic model. The latter are no longer linear functions of Win but still exist regardless of the value of the latter. The mean dislocation density preserves its continuous character.

The situation changes dramatically when nonconvexity is allowed. At positive misfits the energy again consists of curvilinear segments, but they do not intersect each other except for very low values of the substrate modulation W, when the atoms sample in fact the convex part of the potential. As a result, the stability limit fz is no longer defined by the equality of the energies of the commensurate and incommensurate states, but rather by the condition of existence of a solution of the incommensurate state. Note that the stability limit f; at negative misfits is defined as in the harmonic model, as the condition of existence of solutions (eqn. (51)) is always fulfilled. However, at strong enough interfacial bonding the energy of the commensurate state is always greater than that of the incommensurate state. The latter is most probably due to the fact that the energy of the single MDs, which is a decreasing function of the negative misfit (eqn. (53)), becomes negligible in comparison with the energy of the bond distortion. As a result the stability limit f; no longer exists. Another consequence of the distortion of the bonds is the disappearance of the metastability limit at negative misfits beyond a particular value of W. The latter is due to the fact that the distortion of the bonds near the chain ends does not allow the atoms to climb the slopes of the potential troughs. This leads in turn

27

to the conclusion that the activation energy for introduction of MDs at the free ends should be much greater in expanded than compressed chains.

Unlike in the harmonic case, in which the energy 8, of a single dislocation is a function only of the energetic parameters (Z’d = 41, Wlrr, where 1,= (a’/ 2W)‘” [18]), the latter becomes a steep (in fact, exponential) function of the lattice misfit in the nonconvex model (see eqn. (53)). As fi <O, 87d increases with the positive misfit and decreases exponentially with the absolute value of the negative misfit. As discussed above, the latter leads to the disappearance of the negative limit of stability, f;. At zero misfit the energy of a single negative dislocation (heavy wall) is always greater than that of a positive dislocation (light wall), which is due to the asymmetry of the interatomic potential.

The energy of interaction of the MDs also splits with respect to the misfit sign. On the one hand, this is due to the anharmonicity of the pairwise potentials. On the other hand, the distortion of the chemical bonds contributes additionally to the suppression of the interaction between the MDs in expanded chains. This leads in turn to a discontinuous commensurate-incommensurate transition at negative misfits when the substrate modulation W exceeds a critical value such that the interatomic separations sample the non-convex part of the potential. The latter is uncharacteristic of the harmonic model.

3.5. Effect of anharmonicity and nonconvexity in epitaxial growth

In summary, the replacement of the harmonic interactions by more realistic interatomic forces in the 1D Frank-van der Merwe model leads to the following important conclusions concerning the growth of the thin epitaxial films:

(i) Compressed epilayers can crack along the dislocation lines at small values of the positive misfit. Expanded epilayers are expected to crack at the cores of the antisolitons at large absolute values of the negative misfit. In both cases chemical bonds break up in the cores of positive MDs or light walls.

(ii) The limits of stability and metastability of the pseudomorphous state are much greater in absolute value in expanded than in compressed epilayers.

(iii) Thin expanded pseudomorphous overlayers should always be stable beyond some critical interfacial bonding W, irrespective of the absolute value of the natural misfit and should always exist in the metastable state beyond some critical inter-facial bond strength w,, -=X w,.

(iv) The activation energy for introduction of MDs at the free ends is higher in expanded than in compressed films, and therefore expanded films consisting of sep-

arate islands can withstand higher temperatures in the pseudomorphous state than compressed films.

(v) The equilibrium critical thickness for pseudomorphous growth should be much greater for expanded than compressed films.

(vi) The mean MD density should be smaller in expanded than in compressed epilayers at the same film thickness.

(vii) The natural misfit in expanded epilayers is entirely accommodated either by HS or by MDs without an intermediate state.

One of the most important consequences of the nonconvexity of the real interactions, from a technological viewpoint, has to do with crack formation. II- shaped cracks have been observed in compressed Ge films deposited on Si [91]. Cracks have also been found in compressed garnet films grown on garnet substrates [92]. In this case the cracks were observed in slightly rather than strongly compressed samples, in agreement with the predictions of the model. Olsen et al. [93] observed unidirectional cracks in both compressed and expanded epilayers of In,Ga,_,P deposited on (100) GaAs. They found that stretched-out In,Ga, _,P layers cracked at smaller misfits than did compressed layers, in contradiction to the prediction of the above model. Cracks in expanded In,Ga,_& and In,Al,_+4s layers grown on (100) InP by MBE and in expanded In,- Ga, _,As,,P, _-y layers grown on (100) InP by LPE were observed and studied by Franzosi et al. [94]. These authors found that the cracks propagate deeply into the InP substrate. The same phenomenon has also been established for the MDs in MBE-grown In,Ga,_,As/ (100) InP single heterostructures, irrespective of the sign of the natural misfit [95]. The MDs are ‘squeezed’ into the substrate owing to the stress in the overgrowth. This clearly shows the connection between the MDs and the formation of cracks.

4. Interdependence between mode of growth and

interface structure

Both phenomena discussed above are strongly interconnected. On one hand, the mechanism of epitaxial growth depends strongly on the structure and energy of the interface between both crystalline materials. On the other hand, some technologically important aspects of the growing films that are connected with the interface structure, such as the critical thickness for pseudomorphous growth and the mechanism of generation of MDs, are influenced by the morphology of the growing film.

One can conclude, for example, that expanded films with lower dislocation energy have a greater tendency to undergo 2D growth than 3D (Volmer-Weber or Stranski-Krastanov) growth (see eqns. (25)-(27)). The

28

2D-3D transition will take place at higher temperatures when the overlayer is expanded. On the contrary, compressed films such as Ge,Si,_, alloys on Si [37, 381 and In,Ga,_,As on GaAs [96] should exhibit a greater tendency to undergo 3D growth with increasing Ge or In content, respectively. One could expect asymmetry in the 2D-3D transition in the growth of In,Ga,_,As on InP(OO1) when varying the sign of the lattice misfit with alloy composition.

One can draw a definite conclusion concerning the process of growth of compressed epitaxial overlayers that are strongly bound to the substrate from the behaviour of the energy versus island size (Fig. 26). Small monolayer islands are coherent with the substrate. After incorporation of some more adatoms, a dislocation is introduced at the free boundary, but its core bond is stretched out more than the theoretical tensile strength of the material. The overlayer island thus will break up into two smaller islands. This process continues until the density of such small coherent islands becomes large enough. Then they begin to coalesce with each other to produce bigger islands. The gaps shown in Fig. 26 disappear and the overgrowth islands can grow further by incorporation of single adatoms. This process takes place if the misfit is larger than the stability limit fz. If this is not the case, the overlayer islands grow by incorporation of single adatoms and are pseudomorphous with the substrate until complete coverage of the latter is achieved. As the coalescence begins at a later stage of growth, the monolayer film will consist of a large number of small monolayer islands. The adatom concentration on top of the small islands is insufficient to give rise to nucleation of the upper monolayer [16], and hence the formation of the latter will be delayed. Thus layer-by-layer growth will be favoured at positive misfit and strong enough bonding across the interface.

The above conclusions are in good agreement with the recent observations of Becker et al. [97], who reported fragmentation upon high-temperature annealing of 2D Ag islands deposited on Pt(ll1) in the submonolayer region. When deposited at temperatures below 500 K the silver formed large 2D islands pseudomorphous with the substrate. After annealing at higher temperature the silver islands broke down into islands consisting most probably of 7 or 12 atoms. These smaller islands were found to be nearly relaxed. When the deposition was carried out at temperatures higher than 500 K the silver film grew as small islands from the very beginning. The Ag is strongly bound to the Pt and the lattice misfit is positive and large enough (4.3%).

The critical thickness for pseudomorphous growth depends on the mechanism of growth (for a review see ref. 89). It was found that the critical thickness when

the fihn grows as isolated 3D islands is considerably larger than that for infinite monolayers [98]. However, as discussed above, the compressed films have a greater tendency to undergo 3D growth than expanded films. On the other hand, as concluded in Section 3.3, the equilibrium critical thickness for pseudomorphous growth should be much greater for expanded than for compressed films. One could, therefore, expect that compressed 3D islands can become noncoherent at a smaller thickness than can continuous film under the same conditions when anharmonicity is allowed.

The mechanism of generation of MDs also depends on the mode of growth [89, 991. When the film grows as separate 3D islands MDs are introduced in the interface at the free edges [18]. The average strain shows a sawtooth behaviour with the island size when consecutive MDs are generated [loo, 1011. The anharmonicity of the interatomic potential leads to a split of the residual strain with respect to the misfit sign [28]. However, the influence of the nonconvexity is expected to be much stronger. At large absolute values of the negative misfit, when the atoms of the islands sample the nonconvex part of the potential, the interatomic bonds near the free ends should be distorted, as discussed in Section 3.4.2. This results in the disappearance of the metastability limit f&. The atoms do not climb the slopes of the potential troughs and MDs are not introduced spontaneously at the free edges. The islands will grow pseudomorphous with the substrate up to a stage when the coalescence becomes significant. The residual strain should not display a sawtooth behaviour, and the resulting continuous film will be pseudomorphous. However, it will be metastable and MDs should be generated by another mechanism in a later stage [15, 891.

5. Conclusions

We have shown that the criterion of Bauer for the growth mode of thin epitaxial films in terms of the interrelation of specific surface energies is in fact equivalent to the criterion based on the thickness dependence of the chemical potential. The latter, however, is more general in the sense that it determines the direction of the surface transport of adatoms and in turn the transition from 2D to 3D growth mode. It can be concluded that true layer-by-layer growth should be a rare event. It takes place only within no more than two or three monolayers. After that, the growth continues either by formation and growth of separate 3D islands or by simultaneous growth of several monolayers. The critical temperature for the 2D-3D transition cal- culated by considering the kinetics of the thin film growth naturally includes in itself the difference of the

chemical potentials of the consecutive monolayers. It is in good semiquantitative agreement with experimental data for growth of metals and semiconductor materials with covalent chemical bonds.

The one-dimensional non-Hookean models of epitaxial interfaces allow a qualitative explanation of available experimental data and predict some new aspects of epitaxial growth, However, the role of the coupled soliton-antisoliton or multisoliton solutions in expanded chains is still not well understood. It is possible that the latter could explain the formation of cracks in expanded epilayers.

The mechanism of growth of epitaxial films and the structure and energy of the epitaxial interfaces are closely interconnected. Whereas the influence of the non-Hookean properties of the epitaxial interfaces on the growth mode is more or less clear, the effect of the growth mode on the properties of the interfaces is much more complex.

Acknowledgements

The financial support of the National Science Council of the Republic of China is gratefully acknowledged.

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Recent theoretical developments in epitaxy