Structure of epitaxial crystal interfaces

SURFACE

SCIENCE 31(1972) 198-228 0 North-Holland

STRUCTURE

OF EPITAXIAL

Publishing Co.

CRYSTAL

INTERFACES

J. H. VAN DER MERWE* Department of Mathematics and Applied Mathematics, University of South Africa, Pretoria, Republic of South Africa

Theoretical work on the structural influences of misfit, bonding and crystal dimension in epitaxial bicrystals is reviewed. The main contents relate to the model in which the interaction between the crystal halves is represented by a periodic force acting at the interface, the crystals are approximated by elastic continua and the structure is assumed to be governed by lowest energy principles. In this model the misfit is accommodated by sequences of misfit dislocations located at the interface and/or overall lattice strains. The degree to which one or the other mechanism dominates depends on the size and shape of the crystal in addition to the elastic and bonding properties. Realistic approximations for various special cases, e.g. monolayers, thick crystals and intermediate thicknesses, are considered. An exact solution covering the entire spectrum of interest does not exist. Contents

1. Introduction 2. Governing laws 2.1. The crystals 2.2. Tangential interfacial forces 2.3. Normal interfacial forces 2.4. Growth and environment 2.5. Other approaches 2.6. Cases considered 3. A monolayer overgrowth 3.1. The model 3.2. The governing equations and their solutions 3.3. Discussion 4. A semi-infinite overgrowth 4.1. The model 4.2. The governing equations and their solutions 4.3. Discussion 5. Misfit dislocation energy in overgrowths 5.1. Introduction 5.2. The parabolic model 5.3. The model of Ball 5.4. The extrapolation model 5.5. The first approximation model 5.6. Discussion * Now at Department Africa.

of finite thickness

of Physics, University

198

of Pretoria,

Pretoria,

Republic of South

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199

6. Stability of growing epitaxial crystals 6.1. Introduction 6.2. Monolayers 6.3. Thickening overgrowths 7. Concluding remarks

1. Introduction An interface behaves physically, chemically and otherwise, differently from the bulk crystals on either side. Although the differences usually have their origin in features on the atomic scale, they may have considerable influence even on the macroscopic properties of the system. In this review which is concerned with epitaxial crystal interfaces, epitaxy is understood to be the phenomenon in which the orientation of a crystal, grown on a different crystalline substrate, is in a definite relation to that of the underlying crystal. The simplest is a parallel relationship between crystals with similar structures, e.g. Cu in parallel orientation on the (110) face of Ptl). Work on epitaxy has been reviewed by Pashleyz), Chopras), Schneider 4, and others. While observations on epitaxy cover a wide variety of systems, theoretical work has mostly been limited to a few simple cases and to certain special features only. The present review is concerned with theoretical work on the structural influences of interfacial misfit, bonding and overgrowth thickness, in the case of two-dimensional epitaxial overgrowths and on the assumption that the bonding across the interface can be represented by an interfacial potential and that the behavior of the systems is governed by lowest energy principles. Some of the most prominent predictions are: (i) mismatching at the interface is accommodated by misfit dislocations; (ii) when the misfit and the thickness of the overgrowth are small enough the overgrowth is coherent, i.e. the misfit is entirely accommodated by a homogeneous strain in the film; (iii) when the misfit and/or thickness is large enough both homogeneous strain and misfit dislocations are present; (iv) a film whose growth is initially coherent becomes unstable at a critical thickness t, and misfit dislocations are introduced at the free edges of the film; and (v) as the film increases in thickness beyond t, the homogeneous strain decreases. 2. Governing laws 2.1. THE CRYSTALS In the work to be reviewed the interface

is assumed

to be sharp, flat and

J.H.VAN

200

DER MERWE

atomically smooth and the overgrowth to be two-dimensionals). The latter requires the mean diameter of an overgrowth to be very much larger than its height. Surface superstructures revealed by LEEDs), and diffusion modified6) interfaces are disregarded. The atomic interaction within the crystals is approximated on the assumption that the crystals behave like isotropic hookian elastic continua7ps). The dependence of elastic moduli on crystal thickness is neglected. The interfacial atomic meshes are assumed to be rectangular and in parallel orientation. The xy-plane is chosen coincident with the interface and the natural mesh lengths are represented by (a,, u,) for the substrate A and by (b,, by) for the overgrowth crystal B. In this notation the ratio

is referred to as the (natural) misfit. Accordingly a Vernier of mismatch of period ~~=P,b,=(P~+l)a~,

there exists at the interface

i=x-,y,

which is accommodated by two perpendicular tions of spacing pi, as will be seen below.

(2.2)

sequences

of misfit disloca-

2.2. TANGENTIAL,INTERFACIALFORCES At the interface, where atoms are out of register, because of the mismatch, forces due to bonding come into play. This force has a tangential component which tends to pull the atoms on one side into register with their positions of energy minima with respect to the other crystal. This will be loosely referred to as a “register of atoms on either side”. The force will evidently be periodic in the relative tangential displacement UT). It is assumed that this force can be approximated by a force acting between nearest neighbour atoms on either side of the interface. It is conveniently represented in terms of a periodic interfacial potential energy density V. In a first approximation V, for a onedimensional case (misfit along one direction only), has been represented by a Fourier series, cut off after the first harmonic terms-lo) V = j&V,[I - cos(2nUjc)J, where V, is the amplitude model 11, la) :

and

c the period

v = +v; { 1 - cos(2nU/c) in which the second

harmonic

per unit area, or wave length.

+ P2 [l - cos (47rU/c)]),

term has been included

(2.3) A refined

(2.41

was also considered.

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201

The parameter r2 can be varied to obtain different shapes of curves. In particular the crests can be broadened or flattened, if thought proper. The maximum broadening is attained for r2=). The analysis based on (2.4) is vastly more complicated than that based on (2.3). A model which is simpler than the previous two - from the point of view of the analysis - though much less realistic is the one obtained by joining successive parabolic arcs13*14) V = const. V.

(2.5)

The representations (2.3), (2.4) and (2.5) are compared in fig. 1, curves A, (B,, B,) and C respectively. In B, and C, V, and the constant have been chosen such that B, and C osculate with A at the bottom, while in B,, Vd is such that B2 and A have the same overall ampIitude.

DISPLACEMENT

--

Fig. 1. Graphs of the interfaciai potential energy Y {units VO) versus relative atomic dispIa~ment U/LJ.Curve A: representation (2.3); curves B: representation (2.4) for r* -= 4 (BI osculates with A in trough, BZ has same overali variation as A); and curve C: representation (2.5) (osculates with A in trough).

For two-dimensional misfit a two-dimensional representation The simplest representationIs) is considered, namely,

is required.

(2.6) where V, and V, are of the forms given in one of eqs. (2.3), (2.4) and (2.5). It has been shown that in this case the energy associated with the two mismatches, described in terms of two perpendicular sequences of dislocations, may be obtained by addition. The tangential interfacia1 force per unit area, or rather the interfacial shear stress, is obtained from V by simple differentiation: P li = aV/aUi,

i=x,

y.

(2.7)

202

.I. H. VAN

For the representation

DER MERWE

(2.3) this reduces to P zx = 2zd sin (2nU/c),

where the subscript x has been deleted on the right-hand ralization of the well-known Peierls-Nabarro model7). iv,

= pc2/4rr2d

(2.8) side. This is a geneThe amplitude14) (2.9)

has been chosen such that (2.8) satisfies Hooke’s law for small displacements U, ,u being an effective interfacial shear modulus and d the separation of the adjoining atomic planes on either side of the interface. In the past d has been approximated by c li = c (2. IO) V, is evidently related to the strength of bonding between the two substances and hence to the adhesion. According to estimates5,lG) based on simple bond concepts, V, may be as much as one third of the energy of adhesion for short range forces. For long range forces, e.g. van der Waals forces, it is much lesst7). In the predicted results V, appears as an important parameter. 2.3. NORMAL

INTERFACIAL FORCES

The force component normal to the interface also warrants consideration. Contributions from two origins have been dealt with. The one is due to a squeezing action where the atoms ride up each other due to mismatch at the interface. The other contribution is induced by the tangential force when the elastic moduli of the two substances are different7). The latter has been assumed proportionalt4) to the relative normal displacement W of the two adjoining surfaces, i.e. (2.11) pZZ = const. W/d, when expressed in terms of a normal stress. The direct contribution to the energy originating in (2.11) has been shown14) to be less than 8%. Since the inclusion of the normal forces introduces vast complications they are usually ignored in calculations. 2.4. GROWTHANDENVIRONMENT (i) that the crystals are grown under clean conditions, (ii) that the system is so large that it can be treated as a thermodynamic system, and (iii) that the growth is so slow that the system is in quasi-equilibrium all the time. Because of the solid nature of the systems It has been assumedj~Q~14.15)

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considered,

OF EPITAXIAL

the free energy equilibrium

CRYSTAL

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condition

of the epitaxial

203 bicrystal

system reduces to a lowest energy requirement: E = minimum,

(2.12)

where E is the total energy. The additional assumption that the growth is two-dimensional has also been made. The conditions for such growth have been considered previously 5). The energy can be analysed in terms of two parts14,1s-20); the one is associated with the misfit dislocations and the other with a superimposed homogeneous strain. The latter reduces the mismatch and thereby also the number of misfit dislocations. The (free) surface energy dependence on strain 21) is neglected. 2.5. OTHER APPROACHES Models for treating these or similar problems, have been introduced by Brooks22) and Fletcher e5). While the resolution of the mismatch into sequences of misfit dislocations is a natural consequence of the analysis reviewed, Brooks presupposed the suitability of a dislocation description and uses a Volterra type model in his calculations. Fletcher uses a pairwise interaction model between nearest neighbour atoms on either side of the interface. Though this is undoubtedly the most sophisticated and accurate model, its application requires considerable numerical calculations. 2.6. CASES CONSIDERED Even with the grossly simplifying assumptions introduced above the problem of the interaction between the two crystals is very complicated and only for special cases, namely, the monolayer overgrowth and the semiinfinite overgrowth, could it be solved almost exactly. The remaining cases of intermediate thicknesses have been treated approximately by adapting in some plausible way the results of the two extremes. Three-dimensional growth is not dealt with. 3. A monolayer overgrowth 3.1. THE MODEL In the monolayer models4) the atoms are assumed to be embedded in a thin uniform isotropic elastic sheet of thickness t. For a monolayer t is of atomic magnitude 6. With a view to considering the applicability of the results to a film comprising more than one atomic layer, t will not be restricted at this stage. Evidently, these results, when applied to a thick film, will not allow for a variation of strain normal to the film plane.

204

J. H. VAN DER MERWE

The substrate is regarded as rigid. The error introduced via this simplifying assumption, like the one on the strain above, increases rapidly 1s) with increasing

t. By this assumption

the role of the substrate

is reduced

to being

only the source of the interaction potentials (2.3)-(2.5). The letter pair (m, n) is used to enumerate the potential troughs - IZ along the x-axis - as well as the monolayer atoms, starting from a position where an atom and a trough are in register. The positions of the atoms can thus be expressed as &I, =

ax(n + s’“,)

3

Ym = Qy (m +

YI,,)

(3.1)

2

where (QL,, aY q,,) is the position of the (n, m)th atom with respect to the (n, m)th trough. It follows by definition that the linear and shear strains in the film plane are given by24) e, = (5rl+l,m - L, - Pi ‘) ax/b,, e, = (YI,,,+~ - vnm- Pyel) a,lb,, exy = CL, m+ 1

-

L>

ax/b,

+

(YI,+

(3.2) 1, m -

L,)

a,lb,.

The strain can be decomposed into two parts 14, rszo). The one is the elastic relaxation due to bonding and mismatch. It is periodic because it is induced by the periodic interfacial interaction. The other is a superimposed homogeneous strain 2 by which the natural spacing bi is changed to an average value 6,. Because of the periodicity of the former E is also the average of e. The existing mismatch l/Pi and its Vernier period Bi are, in analogy to (2.1) and (2.2)

given by

l/Pi=(r;i-ui)/ai, in the presence

Pi=P&(Pi+l)ai,

of the homogeneous

i=x,L’,

(3.3)

strain

ei = (6, - bi)/bi = (Pi-’ - Pi-‘) rxi,

(3.4)

Eli = Ui/bi.

(3.5)

where

In isotropic elasticity theory, the dependences of the stresses T and the strain energy density E on strain - when the z-dimension is ignored and suitable adjustments are made for the two-dimensional character of the uniform elastic sheet - are of the forms Txy

=

7

(3.6a)

e$] ,

(3.6b)

Ybtexy

and &= &,n

[2 (e: + e,’ + 2ebe,e,)/(I

- cb) +

STRUCTURE

where

Q= tb,b,

OF EPITAXIAL

is the natural

CRYSTAL

overgrowth

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volume

205

per interfacial

atom.

T, is the tensile “stress” in the x-direction and TXYthe “shear” stress in the film plane. The letters ,ub and crb represent respectively the shear modulus and Poisson’s ratio of the overgrowth. The Ti is of the nature of a “line tension” per unit length of a cut perpendicular to the film plane while E is measured in units of energy per interfacial atom. The potential energy of the (n, m)th atom is in accordance with (2.3) and (2.6) given by + +w, (1 - cos 27c)1,,) )

V”, = *w, (1 - cos 27&J

where c, =aX and ~+=a,. The total energy of a configuration is obtained by summation.

(3.7)

of NM atoms

3.2. THE GOVERNING EQUATIONS AND THEIR SOLUTIONS The equilibrium is determined by

position

of the (n, m)th atom 0 =

Only simple solutions

aEp&,,,,

=

aE/dtf,,,

in the interior .

(3.8)

of the form

s’,, = 5,=5(n),

Vl,,=%z=~(~)

(3.9)

are considered. For such solutions the governing conditions are difference equations, are approximated by the differential d2t/dn2

(3.8), which equation91 2s)

= (rc/21:) sin2rrr

for the x-dimension and an analogous meter 1, is given by I: = L&M-K The approximation

of the film

(3.10) is obtained

(3.10)

one for the y-dimension. (1 -

The para-

4.

(3.11)

from expansions

like

5 (n + 1) = 5 (n) + 5’ (n) + $5” (H) + ... , in which higher order derivatives are neglected. The evidently a good one when l(n) varies slowly with n. By simple integration one finds9725) dt/dn

(3.12)

approximation

is

= (1 - ki cos27c~)*/k,l,

(3.13)

= F CL x (;;’ - +)I,

(3.14)

and n%A

where k, is a convenient integration constant, integral of the first kind2G) and n=O at { =+.

F is the incomplete

elliptic

206

J. H. VAN DER MERWE

The dependence of 5 on n is illustrated in fig. 2. In accordance with the concept of dislocations the lines in the positions where 5 = + f, -~_i?,. . are called misfit dislocations. It follows from (3.14) that the dislocations are spaced at regular intervals /i, = PJ, where K is the complete

Fig. 2.

Graph

= 21,k,K

elliptic integral

of relative

By an expansion similar be approximated by

displacement

(3.15)

of the first kind.

< [eq. (3.14)]

on position

n (curve

to (3.12) it follows that the strains

e, = CC,(d?j/dn - PC ‘), Having solved the governing equations to calculate the average tensionzs) TX = [2Wl(l along a cut parallel

(k,) ~,/IT,

-

A).

in (3.2) may

exy = 0.

(3.16)

the results (3.13)-(3.16)

can be used

gdl (%+ Gg

(3.17)

to the y-axis, the work per unit length

Bx = 2/@a, [2a,E (k,)/~k,l,

- a,P,- ’ + o,,i;y]/( 1 - CT&

(3.18)

required to generate (at the free edge) a misfit dislocation lying parallel the y-axis, and the total mean energy per interfacial overgrowth atom + I - k;’ i = x, J’ )

to

1 (3.19)

where E(ki) is the complete elliptic integral of the second kind. The product term in the last line, which represents the direct contribution due to the

STRUCTURE

Poisson

phenomenon,

be inferred

OF EPITAXIAL

vanishes

CRYSTAL

207

INTERFACES

when the average

strains

ei vanish,

as may

from (3.4).

3.3. DISCUSSION In calculating O, in (3.18) it has been assumed that the introduction of an additional dislocation alters the equilibrium conditions only infinitesimally. Also in (3.19) the effects of the free edges have been neglected. These will only be good approximations when the monolayer is large enough. This kind of approximation is characteristic of most of the considerations of the material reviewed. The effects of free boundaries become more important when the overgrowth sizes are small. Some of these effects have been considered by various authors 27$28). An analogous, though one-dimensional model, was analysed previously by Frank and van der Merweg,r5). The results of the two models are qualitatively very similar but quantitatively significantly dissimilar, when applied to a system having two-dimensional imerfacial misfit. This difference has its main source in the Poisson interaction which was not analysed properly in the one-dimensional model. First consideration to this effect was given by Kuhlmann-Wilsdorf and Jesser ls) in their treatment of small epitaxial islands. The problem has also been solved using the refinedr1,12) potential (2.4). The extent to which it alters the predictions will be discussed in section 6. An assumption inherent in the relations (2.2) is that the Pi are integers and consequently that the results are limited to cases where the spacings assume only discrete values in accordance with Ui/bi = P,/(P, + 1) = 9, +, . . . .

(3.20)

Two consequences of this deserve comment. One is that the 6, are larger than the Ui. Ballls) has shown that there is a difference, though small, in the energy calculated for the cases 6,2ai. The other comment is that in applying and displaying the results one usually infers that l/Pi varies continuously, by drawing for example, a continuous curve for the dependence of energy on 1/P. Calculations 13, 23, 24) in which the discrete atomic nature of the system is taken into account with greater precision show that there are cusps in the energy curve in positions where a/b are in the ratio of small integers. 4. A semi-infinite

overgrowth

4.1. THE MODEL For a semi-infinite overgrowth one evidently need not allow for a superimposed homogeneous strain. Since also, the energies associated with the two perpendicular sequences of dislocations are approximately additive for

208

J. H. VAN DER MERWE

an interfacial potential of the kind in eq. (2.6), it is adequate to consider, at this stage, only the misfit in the x-direction. For the latter eq. (2.2) takes the form j?=Pb=(P$_l)a. In the previous case, it was natural to allow for displacement in the monolayer B only and to regard the substrate A as rigid. Thus the problem reduced to finding the equilibrium configuration of the film in the field of the substrate, which evidently has the periodicity a of the substrate. Such a model is clearly not suitable for the semi-infinite overgrowth where both crystals are on the same footing. This difficulty is overcome in an approximate and somewhat artificial way by the introduction of a reference lattice C with spacing c. The lattices of A and B are imagined to be generated from that of C; A in a compression of the span (P+ 1)~ by -]ic and B in an expansion of the span PC by +c yielding*) p=Pb=(P+l)a=(P+.:)c. These equations ratio

determine

(4.1)

the va.lues of p, P and c in terms of a and b. The (4.2)

thus obtained appears to be a more appropriate measure of misfit for the system than l/P as defined in eq. (2.1). By its introduction above, P is an integer. Hence by (2.20) only certain discrete values of u and h are allowed and the comments made previously also apply here. For the purpose of defining the displacement function U it is useful to think in terms of a simple cubic lattice in which the stable positions of interfacial atoms are supposed to be those for which they are in alignment. The description, “corresponding” atoms, will consequently be used to designate an atom pair in which the atoms were originally opposite each other in the reference lattice. The relative displacement CJ of corresponding atoms is thus given by U(x)

= +c + cx/p + Uh(x) - u,(x),

(4.3)

where U, and U, are their tangential elastic displacements when acted on by the tangential interfacial forces (2.8) rx/p is due to the Vernier of misfit introduced by the generation of A and B, while the term $c fixes the origin of coordinates on a dislocation line. The problem of the semi-infinite overgrowth has been solveds) using the form (2.8) namely (4.4)

STRUCTURE

of the tangential

interfacial

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209

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force and neglecting

the normal

forces, the latter

having been justifiedig). Since c depends on a and b the amplitude (2.9) of the interfacial potential will also depend on a and b. It has been assumed14), somewhat arbitrarily that the amplitude itself is independent of misfit, yielding the relation c = d,

PC = pea,

(4.5)

where p. is the value of p when b = a. 4.2. GOVERNING

EQUATIONS AND THEIR SOLUTIONS

For one-dimensional misfit, the present problem becomes one of plane strain in isotropic elasticity. Such problems are most easily analysedzg) in terms of a stress function x satisfying the biharmonic equation Px=O.

(4.6)

The stresses are defined in terms of x by P xx =

while the hookian

pzx= - a2xjax aZ a2XIaZ2, pzz= d2Xja.2, relations

simplify

(4.7)

to

2Pexx = (l - 0) Pxx - CPZZ> and the strain energy density

becomes

c = c(l - g) (P,‘, + PI,) - ~~P,,P,, If the evident

(4.8)

+ 2pL1/4~.

(4.9)

periodicity Pzz(Z, x + P> = Pzz(Z, x)7 . . . 3

(4.10)

Pzx (O> x> = - Pzx (0, - x> >

(4.11)

antisymmetry free boundary 0 = px, (z, x) = . . . etc.

atz=+co,

(4.12)

and continuity Pzz(o~x)=Pzz(o+~x)~

conditions are taken into account integro-differential equation

Pzx(o~x)=Pzx(o+,~),

it follows

that

(4.13)

U satisfies the non-linear

[sin 2nll/ (x + t) - sin 2rt$ (x - t)] cot 31 dt

,

0

(4.14)

210

J. H. VAN DER MERWE

where x = 2nxjy )

p = 2rcdd/~up,

l/G = (I - ~~)~~~ + (1 - ff~)~~~ (4.15) and the subscripts a and b have the evident meanings. Eq. (4.13) has the solutions) $ = u/c,

arctan [(A/?)- ’ tan *Xl,

$(X)=)+x-’

(4.16)

where A=(l The mean interfacial

potential

+pZ)“-fi.

(4.17)

energy density

E, = (/~:*/47t*d) (1 - A)

(4.18)

is obtained by integrating the interfacial potential I/ in eq. (2.3) over the interface. The mean strain energy per unit area of interface

~.. 1_,$e=*” +--(i _cb) (1 _~‘e2H)’

In __.I - A2-

HA2eCZH(H

- 1 + A2eCZH)

where associated with a slab extending from the interface to a distance Iz in crystal B, is obtained by integrating (4.9) over the corresponding volume of B. The total elastic energy in B follows by letting h-t co in (4.19) : Et = - I( 1 - CT&c2~~~/4~2~~d~ III (1 - A*).

(4.20)

The total elastic energy in the system is accordingly E, = - (pc2fl/4n2d) and the total energy

In (I - A2),

per unit area of the bicrystal,

the so-called

(4.21) interfacial

energy, E = E, + E, = (~c2/4n2d) Note that the energies of the interface.

[i - A - /I in (I - A’}].

in eqs. (4.18)-(4.22)

(4.22)

are all mean values per unit area

4.3. DISCUSSION A graph of U(x) again displayss) the dislocation character of the interfacial atomic configuration. The resolution of the interface into a sequence of dislocations therefore follows naturally from the analysis. The interfacial energy E, is apart from being proportional to the ampIitude V0 =pc2/4zZd of the interfacial potential (2.3), determined only by the parameter /!J which, by (4.15), is a simple function of the prominent parameters

STRUCTURE

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CRYSTAL

211

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p, c, p,, pLb,a, b of the system. A graph of E(P) is given in fig. 3, curve C. Curves A and B illustrate the dependences of E,,, and E, on p. The validity of Hooke’s law in crystals A and B is a basic assumption of the calculations. The model predicts, in the extreme case of a single dislocation, a maximum strain of approximately 22% at the dislocation which is

I

0 0.1

05

1.0

I

I

I

I

I

1.5

I

I

I

1.8

a-

Fig. 3. Graphs of the mean interfacial energy E per unit area [eq. (4.22)] in units of PC/~+ versus the parameter B [eq. (4.191 for quadratic symmetry (ax = uy, . ..). P~ = fib = ~CO and u = f. Curves A, B and C represent Ee, Em and E, respectively.

certainly far beyond the limit below which Hooke’s law is satisfied. The strain decreases to less than 3% in a distance of approximately three atomic spacings. The exact error which this super critical strain introduces in the energy has not been estimated. It is small but probably not negligible. A criticism which is also related to the strain is the fact that it varies so rapidly in the immediate neighbourhood of the dislocation. The continuum approximation certainly breaks down there. Because of the small spatial extent of this region it is suggested that the error is small. The energy associated with that part of the crystal B, extending outwards from the interface and commencing at a distance h from it, is given by R(h)=E,b-E:(h)

(4.23)

as may be inferred from (4.19) and (4.20). By simple calculations it follows that R(h)t2% of E,” for h =+p. This shows that practically all the energy

212

I. H.

VAN DER MERWE

resides in a slab of material extending to distances +p on either side of the interface. This result justifies the application of the relations (4.1 S), (4.19) and (4.12) to films of finite thickness t provided t3:y.

(4.24)

Stated differently, the condition (4.24) is a useful criterion for the applicability of the predictions of the semi-infinite model to actual systems, which are naturally of finite dimensions. An aspect of the model which should rightly be commented on is the implicit assumption that the misfit dislocations are everywhere regularly spaced. The spacing will evidently be influenced by the presence of free surfaces, in particular surfaces where the interface boundary emerges. By Saint Venaint’s principle the average effect will be small when the interface contains many dislocations. Concluding this section, mention should be made of attempts to solve the problem using the refined interfacial potential (2.4). Three approximations have been obtained. The itterative approximation converges only in a limited and un-important region of misfit, but is certainly the most accurate in that region. The variational approximation is complicated, but certainly the most accurate, when averaged over the entire region of interest. The third one, the so-called first approximation, is by far the simplest and also realistic, over the entire range of interest. It assumes that the solution (4.16) is a useful first approximation of the problem using the refined potential (2.4). Since the interfacial displacements are the same, the strain energy will again be given by (4.21). The mean interfacial energy is approximated by substitution of (4.16) into (2.4) and by performing the necessary integration. The following results are obtained 12) E:, = (pc2/8n2d)

[l + ($p - 1) A],

E = (pc2/47r2d) (4 [ 1 + (38 - 1) A] - j3 In (1 - A’)}.

(4.25) (4.26)

In these results VA of (2.4) has been chosen such that the corresponding shear stress (2.7) satisfies Hooke’s relation for small displacements. (See curve B, in fig. 1.) Also the substitution r2 = B has been made. 5. Misfit dislocation energy in overgrowths of finite thickness 5.1. INTRODUCTION Two extreme cases of epitaxial bicrystal systems, one comprising of a monolayer overgrowth and the other a semi-infinite overgrowth, have been dealt with above. Certainly of great interest is the wide range of intermediate cases representing the two-dimensional growth of epitaxial crystals.

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OF EPITAXIAL

CRYSTAL

213

INTERFACES

Since the energies associated with a superimposed homogeneous strain and with the mismatch - misfit dislocations - are additive, they may be analysed separately, the latter in this section. It has already been indicated, with reference to relation (4.24) that the misfit dislocation energy Ed of an overgrowth of finite thickness t can be approximated by that of a semi-infinite overgrowth when the condition (4.24), namely t 2 4P (5.1) holds true. This evidently does not cover the case of a film comprising a few atomic layers if its mismatch is small and its dislocation spacing p is large. Neither is this region covered by an extrapolation of the results obtained for a monolayer, by putting t = nb,, (5.2) where n is the number of atomic layers and b, their spacing. The reason is that the monolayer model does not allow for proper elastic relaxation in both the multiple layer and the substrate. The error introduced by assuming the substrate to be rigid is, as will be shown below, already significant for a monolayer and increases rapidly with increasing thickness. Recall that, in addition, the model restricts the strain in the film to having no variation normal to the interface. In the sections below, four attempts to find useful approximate expressions for the misfit dislocation energy, when the overgrowth is of finite thickness, will be described and compared. The models used in these attempts are referred to as the parabolic model, the model of Ball, the extrapolation model and the first-approximation model. 5.2. THE PARABOLICMODEL In this modeli4), in which the interfacial potential is represented by a succession of parabolic arcs (2.9, the shear stress (2.7) takes the form Pa (xl = CICJ(x)/d 9 Additional

accuracy

is achieved Pzz

-Jc<

by including

u<+c.

(5.3)

the elastic normal

(xl = 2PW(x)/N1 - 20)4

stress (5.4)

introduced in eq. (2.11). W(X) is th e relative normal displacement of atomic planes adjoining the interface. It is also more convenient in the analysis here to use the relative tangential displacement in the form

u (x) = cx/p+ u,,(x) rather than dislocations.

that

in (4.3). In (5.5) the origin

u,(x) is chosen

(5.5) midway

between

214

J. H.VAN

DER MERWE

The method, formulated in terms of the relations (4.6)-(4.13) is also applicable to this problem. The difference is that instead of the interfacial force (4.4), the system is subjected to the interfacial forces (5.3) and (5.4). Because of the linearity of these in CJ and W, the governing eq. (4.14) is replaced by a set of six linear algebraic equations determining six Fourier coefficients. By the methods described above the misfit dislocation energy E,, has been calculated, yielding the result 14)

(5.6) where Bi and C,” are two of the said Fourier coefficients. Since, firstly, the Fourier coefficients are complicated functions, secondly, the series (5.6) converges very slowly for the important region of small mismatch and thirdly, the parabolic model is unrealistic anyhow, there seems to be no point in giving further details here. One feature of the results seems worth mentioning however, and that is that except at very small misfits, the energy, even for a monolayer, differs relatively little from that for an infinite crystal. This result is illustrated by curves A and B in fig. 4 where Ed is shown as a function of the misfit l/P = (b - 0)/u.

0

Lo

.05

.lO MISFIT

15 l/P

-

Fig. 4. Curves A and B represent the mean misfit dislocation units of pc/4nz versus one-dimensional misfit l/P = (b - a)/a for and of monolayer thicknesses, respectively. The graphs have .LI~= fib = PO and o = $. Curve C is obtained from

energy Ed [eq. (5.6)] in overgrowths of infinite been calculated taking eq. (4.22).

STRUCTURE

OF EPITAXIAL

CRYSTAL

INTERFACES

215

5.3. THEMODELOFBALL~*) This model also uses the parabolic representation in eq. (5.3) but achieves appreciable simplification, in comparison with the previous results by ignoring the normal force (5.4) and the strain gradient perpendicular to the film plane. The deletion of the normal force has been justified previouslyl~), while the strain gradient is evidently negligible for a very thin film. That the latter must be important is illustrated by the present calculations which show that even for a monolayer, the energy with an elastic substrate is significantly less than on a rigid substrate. Ball correctsi*) approximately for the overestimation of energy due to the unrealistic character of the parabolic interfacial potential (see fig. I) by introducing a scale factor z. A scaIed interfacial shear modulus p* = /Cr

(5.7)

replaces JXin (5.3), in such a way that the slope of the energy curve for a monolayer, at zero misfit, matches that of the more realistic sinusoidal models (2.3) or (2.4). The implicit assumption is made that the same scale factor yields reasonable matching at larger misfits and for thicker overgrowths. The fact that the curves yielding I”* differ very little up to misfits of 20%, is regarded as evidence in support of the assumption. The curves are compared in fig_ 5, B and A corresponding to representation (2.3) and the scaled curve respectively.

Fig. 5. The mean misfit dislocation energy Ed (units ,@2rP) per unit area versus misfit c/p [eq.(4.2)] for a monolayer overgrowth, assuming quadratic symmetry (as =Q,, . ..). pea= pb =po and o = f. Curve A: scaled energy E*d [eq. (5.1 I)] for rigid substrate; curve B: sinusoidal model (2.3) for rigid substrate (obtained from (3.19) for zero average strains a); curve C: scaled energy .!?*d [eq. (5.1 l)] for elastic substrate.

216

J. H.

It has been shownls)

VAN DER MERWE

that z = 64/x4

for (2.3))

f5.8)

and z = 8 [2” + In (2* + 1)]*/rc4

for (2.4) with r2 = t.

The tangential interfacial shear stress p,,(O, to the displacement ub in the film by

x) is assumed

(5.9) to be relate

pzx (0, X) = 2/.+,(1 + CT*)t d2z+,/dx2.

(5.10)

The substrate is dealt with according to the methods of section 5.2. The governing equations are simple algebraic equations for the unknown Fourier coefficients. By the usual formalism one obtains for the scaled mean interfacial energy per unit area E* the relation m 1 E*d = ‘:.f n2 +np-” 4n2 c ll=l

where P = 27W,lrP*P

(1 - @Al 3

$T’

y = P*y2/[47r2P* (I + Oh) tc] .

(5.11)

(5.12)

The series (5.11) can also be expressed in terms of digamma functions. Various special cases flow from (5.12) and (5.11): (i) If either pb or t is infinite y vanishes and the system is equivalent to a pair of which the member B is rigid. (ii) If p, is infinite the system represents a film of thickness t on a rigid substrate. (iii) If in addition t=b, we have a monolayer on a rigid substrate. (iv) If pa is finite and t = b, we have a monolayer on an elastic substrate. Cases (iii) and (iv) have been used to obtain curves B and C in fig. 5. The additional relaxation introduced by allowing for elastic relaxation in the substrate is seen to be significant even in the case of a monolayer. 5.4. THE EXTRAPOLATION MODELS()) In dislocation theory it is often assumed, with some justification, that the field of a dislocation is screened by that of others and/or by the presence of free surfaces. The screening is approximately effective in regions which are further away from the dislocation under consideration than half of the mean dislocation spacing, y, or the mean distance of the dislocation from the free surfaces. Hence if t be the thickness of the overgrowth, the effective range $4 of the field of a misfit dislocation is defined by (5.13)

STRUCTURE

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217

INTERFACES

or more conveniently, cfg = cjzt = c/p

c/p Q c/2t c/p > c/zt *

(5.14)

The relations (5.14) are illustrated by the straight line sections AB and BC in fig. 6. The considerations related to eq. (4.24) and to section 5.2 suggest that the approximation represented by CB extends downwards, or stated differently, that the accuracy of the representation can be improved by introducing a gradual transition, using for example the simple parabolic arc DB,

Fig. 6. Dislocation field range parameter Curve ABC: the usual approximation;

c/q [eq. (.5.14)] versus misfit C/P [eq. (4.31. curve DBC: the refined approximation.

rather than the straight line section AB. The refined representation cl4 = c/4t + tc/p2

= ClP

c/p < c/2t c/p $ c/2t.

is given

(5.15)

For a system having quadratic interfacial symmetry this approximation yields the expression (5.16) with F(P) = E 0Wc2/47t2~)

(5.17)

for the mean interfacial energy per unit area; where E@) is given by either (4.22) or (4.26).

218

J. H. VAN DER MERWE

5.5. THE FIRST APPROXIMATIONMODEL30) This analysis, like that of the so-called first approximation associated with eqs. (4.25) and (4.26), uses U(x) = cqir(x), defined in (4_16), as an approximation of the displacement function. The merit of this method is that it allows for unrestricted, though approximate, elastic relaxation in the crystals. By the approximation adopted complicated governing equations are essentially replaced by the simple expansion *l Ann- I sin mx,

ub (x) - I,I, (x) = :

m = Zscnjp,

(5.18)

c 1 in which the coefficient A is known to be given by (4.17). As in previous cases the stress function method leads to a set of simple linear equations from which the unknown Fourier coefficients can be expressed in terms of A. The approximate elastic energy E, in the crystals is most easily calculated as the work done by the interfacial stress in the elastic displacements of the two adjoining surfaces. One finds x

E, = ;;,

(s’ - m2t2)A"'/n (s2 - m2t2)ib/iP + (SC- ml) i/A,

p n=

(5.19)

I

where *-I A

= iL, ’ -I- i; ’ ,

mt==2nnt/p,

ki = pi/( 1 - fli),

s=sinhmt,

i=a,b,

c=coshmt.

This method certainly underestimates the elastic energy, in particular for very thin films. The approximate mean potential energy of mismatch Em is obtained by substituting (4.16) into the potential energy density functions (2.3) or (2.4), to obtain the results (4.22) or (4.26). The variation of the interfacial energy E, = E, + E, is illustrated

(5.20)

in fig. 7.

5.6. DISCUSSION Approximate models, introduced to describe the misfit dislocation energy of a system with an epitaxiai film of finite thickness f, have been analysed above. The results of the last three models are compared in figs. 7a and 7b, for a monolayer and a ten-fold layer respectively, The curves have been calculated taking pill=g,=po and ca=gb=c=+. In each figure curves A, B and C correspond respectively to the model of Ball, the first approximation and the extrapolation model.

STRUCTURE

ID

OF EPITAXIAL

CRYSTAL

INTERFACES

219

r

(a)

(b) Fig. 7. Misfit dislocation energy & (units /~/2n~) versus misfit c/p [eq. (4.2)1 for quadratic symmetry (az = ay, . . .), pea=/ID = ~0 and (I = 3. Curves A, B and C represent model of Ball [eq. (5.1 l)] extrapolation model [eq. (5.16)1 and first-approximation [eq. (5.19)]. respectively. (a) Monolayers; (b) five-fold layers. Curve D represents in both cases a thick (semi-infinite) overgrowth obtained from eq. (4.22). The model of Ball is certainly the best approximation for a monolayer, since it represents most closely the properties of the model i.e. a film with constant normal strain gradient on an elastic substrate. The artificial scaling cannot be too great an objection since the scaled curve matches closely (fig. 5, curve A) the corresponding curve of the realistic sinusoidal model. In comparison, B lies somewhat too low, while C lies grossly too low (fig. 7a). The main reason why B is too low is that it underestimates the strain energy.

J. H. VAN DER MERWE

220

6. Stability of growing epitaxial crystals 6.1. INTRODUCTION This section reviews the stability the supposition is a minimum,

of growing

that the stable configurations i.e. eq. (2.12), namely

epitaxial

films, predicted

on

are those for which the energy

E = minimum (6.1) is valid. It has been shown above that mismatch in an epitaxial bicrystal is accommodated by misfit dislocations. If a homogeneous strain, which reduces the mismatch, is superimposed on the periodic strains of the dislocations the number of dislocations and consequently the energy associated with them, will be reduced. However, positive energy of homogeneous strain is introduced at the same time. Approximately, the former varies linearly with strain and the latter quadratically. Hence there always exists an energy minimum. Some attempts to calculate the minimizing strain will be considered below. The increase and decrease of homogeneous strain is accompanied by the emission and generation of dislocations respectively, when b > a. These processes will require activation energies depending on the mismatch and strain. Accordingly there also exist metastable configurations 15$I*). Only the simple case comprising pure misfit dislocations will be dealt with. Thereby the important case of imperfect misfit dislocations z”) with slip vectors which are inclined to the interface and which move onto oblique slip planes are not covered. Pure misfit dislocations generate or escape at the junction of the interface and the free overgrowth edges along the perimeter of the overgrowth. The associated problem in the case of overgrowths comprising more than one atomic layer is complicated and unsolved as yet. The stability and metastability

of monolayers

briefly reviewed

have been analysed

fairly satisfactorily

and will be

first.

6.2. MONOLAYERS~~~~~) The condition that an island overgrowth x = + +L, and y= ++L, be in equilibrium absence of external forces is that TX=0 TY=O

atx=FfL,, aty=i+L,,

with rectangular boundaries at (stable or metastable) in the

(6.2)

where the tensions T are defined in section 3. Although these conditions cannot be satisfied identically for the simple kind of solutions (3.9), these can be used to show that such a configuration will be in equilibrium, except

STRUCTURE

OF EPITAXIAL

CRYSTAL

1NTERFACES

possibly for small regions at the corners of the rectangle, I/P, satisfies an inequality which simplifies to D/(1 f

5)

E <

l/P < l/(1 -i- C) E,

221

when the misfit

(6.3)

for a coherent configuration of a system with quadratic (a, = a,,, . . . , 1, = I,,, . . .) symmetry. If the misfits in the two directions were equal in magnitude but oppostie in sign, we have 0 < l/P < 1/t

(6.4)

instead of (6.3). When there is one-dimensional misfit only, (6.3) is replaced by (6.4). The equilibrium configurations determined above, include among them the stable ones. For a monolayer with a fixed number of atoms the condition (6.1) will be satisfied when the mean energy per atom 8, defined in eq. (3.19), is minimized, i.e. when 0 = aqae, = a.qai;, . (6.5) The conditions

are equivalent

to 0 = iii, = SC?,

(6.6)

where the mean work ti is given in eq. (3. IS). It accordingly follows that the condition for stable equilibrium is given by an equation of the form 1/P, = 2E (k,)/nk,E, for the x-direction to

and an analogous

-f- cr~Jcl,

one for they-direction.

l/P, = 2/7cl(l + rr)

(6.7) Eq. (6.7) reduces (6.8)

to define the limiting misfit l/P, at which a coherent configuration, in the case of quadratic symmetry, becomes unstable. Frank and van der Merwer5) have estimated - for a one-dimensional model - that the limiting misfits l/P, herent monolayer loses stability and imately 9% and 14%. The limiting model discussed above are related to

=2/n!, and l/P, = l/Z, at which a cometastability respectively, are approxmisfits l/P, for the two-dimensional l/P, by

I/P, = [(I - c>i(l +

41”WJ.

(6.9)

The difference is seen to be due to the Poisson phenomenon and to infer a reduction of approximately 25% in the limiting misfits if B is taken equal to 3. The dependence of relative mismatch I/P on relative misfit i/P at which the monolayer system is stable, is ilIustrated by curve A in fig. 8; curve B

222

.I. H. VAN DER MERWE

Fig. 8. Graph of relative mismatch l/P versus relative misfit l/P for which monolayer is stable; OC and curve A being for relation (6.7) and OD and curve B for the Frank and van der Merwe model.

being predicted by the one-dimensional model. Along OC the mismatch I/P is zero showing that for misfits below those defined by eq. (6.8) the coherent configuration is stable and the misfit is entirely accommodated by homogeneous strain. As the relative misfit increases beyond the point C, the relative mismatch increases very rapidly by a process in which misfit dislocations are fed into the interface at the free edges. Theoretically, the misfit will always be partially accommodated by homogeneous strain as is manifested by the fact that l/P is always less than l/P. An important result of the one-dimensional calculations, apart from the difference in the limiting misfits defined by points C and D is that in this model the limiting misfit is critical by having a vertical slope at D, predicting that there are either no dislocations or many15). If the properties of an n-fold layer are predicted from those of a monolayer by the substitution t=nb, in eq. (5.2) it follows that l/PM = n-+/p,,

(6. IO)

where P, is the corresponding limiting misfit for an n-fold layer. It has been shown in section 5.3 that this kind of approximation becomes progressively poorer as n increases because the model does not allow properly for elastic relaxation in both the film and the substrate. The approximation is already rather crude for n = 2.

STRUCTURE

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CRYSTAL

223

INTERFACES

6.3. THICKENING OVERGROWTHS The total energy E, of a system comprising an epitaxial overgrowth of finite thickness t on a thick (semi-infinite) substrate may be written in the form l*) E, = E,S,

(6.1 I)

+ (E;r + E:) S + E, + E,, ,

where S, is the interfacial area when the film strain d vanishes, S the area when C?is not zero, EH the energy of homogeneous strain per unit interfacial area when C#O, Ed the average misfit dislocation energy per unit interfacial area for misfit in the i-direction (i=x, JI) under the existing strain C, Er the total free energy of the free surface of the overgrowth and Ead the total adhesive energy for the coherent configuration. Ead is, by definition, independent of strain. Ef is supposed to be independent of strain, thereby ignoring the strain dependence predicted by Drechsler and Nicholaszl). The minimization of E, is therefore equivalent to the minimization of ETS, = E,S,

+ (E: + Ei) S ,

or ET=EH(e-)+2E,(P)(l for the simple case of quadratic

interfacial

S=S,(l

(6.12)

+C)‘, symmetry

for which (6.13)

+c)2,

E, = 2~, (I + CT&te2/( 1 - crb).

(6.14)

For Ed one may choose one of the expressions in eqs. (5.11) (5.20) depending on the accuracy required. The energy ET is minimized for a strain 2, defined by

aE,laz=o,

e=c m.

(5.16) and

(6.15)

In writing down the expressions for E,, it must be borne in mind that the value of Ed is needed for which the average strain e = (6 - b)/b

(6.16)

is not zero. This requires that the relations of the kind evaluate Ed, must be replaced by the analogous ones p=I%=(P+

(4.1), needed

l)a=(P+i)C?.

to

(6.17)

The relationsrs) e, = (Cl - b)/b)

P=/&(l

p = c/p = - e,i( 1 + &e,) ,

++P)y

jj = C/p = (a - e,)/[ I + f (? + e,)]

easily follow from (4. l), (4.5), (6.16) and (6.17)

(6.18)

where e, is the strain in the

J.H.VAN

224

DER MERWE

coherent (6=a) configuration and /, and p may be regarded as useful measures of misfit in view of the reIation (4.2). From the relations above (6.15) is equivalent to an equation of the form (6.19) from which one may solve fof(eO’

em7 r) = ”

% =P(ea,

(6.20)

t),

and calcuIate E T,

min =

ET

teOy

q

(e0,

t)y

fl

.

(6.21)

The dependence

of ET, min on e0 or p for different values of t or of ET, min on t for different values of e, or p may accordingly be calculated. Of particular interest is the limiting condition E;, = e, = f?c

for which the coherent

configuration,

(6.22)

defined by i? = e,

(6.23)

becomes unstable. One may then either solve (6.19) for t in terms of e, to find the limiting thickness (6.24) t, = t(eo) = t’(p) for which the coherent configuration or solve for the limiting misfit

with given misfit p becomes

unstable,

(6.25) PC = P(f) in terms of the thickness. BallIs) has calculated the relation between C, and t in (6.20) for the case of Cu on Ag using the expression (5.11) for Ed in the equilibrium equation (6.12). The corresponding misfit of - 13% is so large that a coherent overgrowth is unstable, even for a monolayer. When Zm is known the corresponding misfit dislocation spacing Is, can be obtained from eqs. (6.16)(6.18). Ball accordingly compared his predictions of p, with the observations of Gradmannar) and with predictions of the simple model of Frank and van der Merwe15). Curve A (fig. 9), for Ball’s model, and curve B, for the simple model, have been constructed from the curves of Ball and are not accurate. Curve A is seen to approach the zero limit more slowly than curve B. Because the first approximation model surpasses that of Ball in the representation of the elastic strain in the overgrowth, a curve based on expression (5.19) will approach the zero limit even more slowly. The dependence of the limiting thickness t, on coherent strain e,, has also been calculated by Ballrs) from his model. The result is illustrated in fig. 10 for an average case in which ~in=~r,=~, and 0=0.3.

STRUCTURE

OF EPITAXIAL

CRYSTAL

225

INTERFACES

.02

.Ol

0

Fig. 9. Dependence of equilibrium strain dm on overgrowth thickness t in eq. (6.20) (units of a) for Cu on Ag. Curve A: model of Ball, curve B: Simple model of Frank and van der Merwe.

15 -

5-

0

Fig. 10.

I .Ol

I

.02

STRAIN

.03

/e,]

I

.04

I

.05

Dependence of limiting thickness tc (at which coherent overgrowth unstable) on the magnitude le0 of the coherent strain [see eq. (6.24)].

7. Concluding remarks The influence of bonding, misfit and overgrowth thickness on the structure of epitaxial crystals, grown under equilibrium conditions, has been discussed. The discussions were restricted to cases where the extent of the interface, when

226

J. H. VAN

DER MERWE

misfit dislocations are present, is large compared to either the spacing of the dislocations or the range of their interaction. The case of three-dimensional overgrowths, where the diameter and height of the overgrowth are of the same order, as well as the case of small islands, where edge effects are important and the introduction of a single dislocation alters the configuration drastically, have not been considered. Also, the predictions reported, have been deduced using a simple interfacial potential model. In short, the more interesting cases and possibly more versatile models, were not discussed. Acknowledgements The author wishes to thank the South African Council for Industrial Research for an International Conference Grant and the University of South Africa for both a grant and special leave. Also, the work was partially supported by the Center for Advanced Studies at the University of Virginia. References I) G. I. Finch and C. H. van Sun, Trans. Faraday Sot. 32 (1936) 852. 2) D. W. Pashley, Advan. Phys. 14 (1965) 327. 3) K. L. Chopra, Phys. Status Solidi 32 (1969) 489. 4) H. G. Schneider, Ed., Epiraxy and Endotaxy (VEB Deutscher Verlag fur Grundstoffindustrie, Leipzig, 1969). 5) J. H. van der Merwe, Basic Problems, in : Thin Film Physics, Eds. R. Niedermayer and H. Mayer (Vandenhoeck and Ruprecht, Gottingen, 1966) p. 122. 6) J. S. Vermaak and J. H. van der Merwe, Phil. Mag. 10 (1964) 785; 12 (1965) 453. 7) F. R. N. Nabarro, Proc. Phys. Sot. (London) 59 (1947) 256. 8) J. H. van der Merwe, Proc. Phys. Sot. (London) A63 (1950) 616. 9) F. C. Frank and J. H. van der Merwe, Proc. Roy. Sot. (London) Al98 (1949) 205. 10) J. Frenkel and T. Kontorova, Physik, 2. Sowjetunion 13 (1938) 1. 11) F. C. Frank and J. H. van der Merwe, Proc. Roy. Sot. (London) A200 (1949) 125. 12) C. A. B. Ball and J. H. van der Merwe, Phys. Status Solidi 38 (1970) 335. 13) J. C. du Plessis and J. H. van der Merwe, Phil. Mag. 11 (1965) 43. 14) J. H. van der Merwe, J. Appl. Phys. 34 (1963) 117, 123. 15) F. C. Frank and J. H. van der Merwe, Proc. Roy. Sot. (London) Al98 (1949) 216. 16) G. Gafner, D.Sc. thesis, Univ. of Pretoria (1960) p. 68. 17) J. K. Mackenzie, Proc. Phys. Sot. (London) A63 (1950) 1370. 18) C. A. B. Ball, Phys. Status Solidi 42 (1970) 357. 19) W. A. Jesser and D. Kuhlmann-Wilsdorf, Phys. Status Solidi 19 (1967) 95. 20) J. W. Matthews and J. L. Crawford, IBM Research Report RC 2748, Yorktown Heights, N.Y., 1970. 21) M. Drechsler and J. Nicholas, Chem. Phys. Solids 28 (1969) 20-64. 22) H. Brooks, Metal Inceufaws (Am. Sot. Metals, Cleveland, Ohio, 1952) pp. 2064. 23) N. H. Fletcher, J. Appl. Phys. 35 (1964) 234. 24) N. H. Fletcher and P. H. Adamson, Phil. Mag. 14 (1966) 99. 25) J. H. van der Merwe, J. Appl. Phys. 41 (1970) 4725. 26) Jahnke-Emde, Funktionentafeln (1938) pp. 52-70. 27) R. Vincent, Phil. Mag. 19 (1969) 1127. 28) W. A. Jesser and J. H. van der Merwe, Phil. Mag. 24 (1971) 295.

STRUCTURE OFEPITAXIAL CRYSTALINTERFACES

227

29) S. Timoshenko, Theory of Elasticity (McGraw-Hill, New York, 1934). 30) J. H. van der Merwe and N. G. van der Berg, Surface Sci. 32 (1972) 1. 31) U. Gradmann, Physik Kondens. Materie 3 (1964) 91.

Discussion R. DE WIT (National Bureau of Standars, Washington, D.C.): Is there any way to experimentally find out which of the models is most accurate? J. H. VAN DER MERWE: Matthews has done some experimental work which is relevant.

Hopefully

he will deal with it himself.

J. H. MATTHEWS (IBM York) : The accuracy

Watson

which of the calculations mentions

Center,

Yorktown

Heights,

New

is at present too low for one to determine

is the most precise.

later, discrepancies

as those described

Research

of experiments

However,

as van der Merwe

exist between all the equilibrium

by van der Merwe)

and experiment.

arise, or seem to arise, from difficulties

associated

theories

(such

These discrepancies

with the creation

of misfit

dislocations. H. B. AARON (Ford test the theories

Motor

Co.,

one obtains

the crystal thickness,

are the relations

Michigan):

How would one

experimentally?

VAN DER MERWE: Theoretically elastic strain,

Dearborn,

relations

between the average

the misfit and other parameters.

one would test experimentally.

These

It is found that the predic-

tions of the theory are rather low (i.e. predict a lower elastic strain) than is observed,

particularly

for large thicknesses.

ANONYMOUS: You said Ball’s model is good for thin crystals. define thick and thin crystals.

How do you

What is their order of magnitude?

VAN DER MERWE: One would expect the model of Ball to be useful for a double or even a three fold layer but to be rather inaccurate ANONYMOUS: How do the results compare

of the periodic

for thicker films.

potential

calculations

with those that use pairwise interactions?

VAN DER MERWE: You

mean from

the point

of view of calculating

the

energy for example. ANONYMOUS: Yes. VAN

DER MERWE:

tions of Fletcher. by making

suitable

I have only attempted

Agreement choices

for the relevant

ANONYMOUS: Dr. Fletcher tials which were not periodic VAN

a comparison

to within approximately parameters.

has used coincidence

[N. H. Fletcher,

Phil. Mag. 16 (1967)

G. BISHOP (Army Materials

lattices with some poten-

potentials.

DER MERWE: He used a pairwise interaction

culations

with the calcula-

20% can be obtained

and Mechanics

model for his 1967 cal-

1591. Research

Center,

Watertown,

228

Massachusetts) non-rectangular

J. H. VAN

DER MERWE

: Has there been any treatment of epitaxy in which you treat cells in which there is a disregistry in the angle a between

the a and b vectors defining the cell. VAN DER MERWE: In his 1967 paper Fletcher has dealt with such a case. If there is only an angular misfit the interface can be analyzed according to the present considerations in terms of screw dislocations.