Full text of DOI:10.1007/s11661-002-0370-4

Reciprocal-Space Formulation and Prediction of Misfit
Accommodation in Rigid and Strained Epitaxial Systems
MAX W.H. BRAUN and JAN H VAN DER MERWE
Geometrical properties form a key aspect of any description of heteroepitaxial systems in which
orientation, symmetry, and lattice parameters differ. An energetically founded epitaxial criterion,
which is geometric in nature, for matching at a planar interface is derived from a generalization of
the Frank–van der Merwe theory and the rigid models introduced by Reiss and van der Merwe. The
criterion is most naturally formulated in reciprocal space as the matching of overgrowth and substrate
reciprocal lattice vectors and is visualized with a construction analogous to the Ewald construction.
Structure factors are introduced and account for rigid translations and the nonprimitive nature of
substrate and overgrowth surface unit cells. This article focuses on the derivation of the epitaxial
criterion and its consequences as a basis of a description of epitaxial configurations, pseudomorphism,
and the parameters of dislocation or misfit vernier arrays, in terms of crystallographic conventions.
The strength of the description is its general nature, as it is general and applicable to any combination
of crystal symmetries or mismatch and can be used to predict, or interpret, interfacial structure.
I. INTRODUCTION
Intuitively, the geometries of crystal surfaces on either
side of a planar epitaxial interface can be described very
generally in terms of surface-reciprocal lattices, and it may
be expected that an epitaxial criterion related to the matching
of rows of atoms on either side of the interface can be
conveniently expressed in terms of these. Here, we derive
such a matching criterion from energy considerations.[2] The
interfacial energy is minimized in an epitaxial system
described with a general, but rigid and essentially atomistic,
model.[3,4] The criterion is made particularly useable through
the introduction of a geometrical construction analogous to
the Ewald construction, common in crystallography.
Consequences of the criterion are discussed, including
prediction of ideal epitaxial configurations,[3] and this article
introduces useful measures of interfacial misfit related to
both direct and reciprocal space. Misfit is accommodated in
several ways in real systems: homogeneous strain to allow
exact matching of the interface structures (pseudomor-
phism), both one- or two-dimensionally; periodic relaxation
characteristic of arrays of misfit dislocations; or misfit ver-
nier. From the epitaxial criterion we derive a formulation
for misfit dislocation spacings, directions and Burgers vec-
tors, and process for the unique characterization of misfit
dislocation arrays in terms of their degree of edge and screw
character. A real system may be expected to show combina-
tions of these misfit accommodation mechanisms, and their
prediction is dealt with elsewhere, where the elastic proper-
ties of the systems are taken into account.
A
PPLICATIONS of epitaxial systems range from met-
allurgical materials to semiconductor electronics, optics, and
superconducting thin-film devices. Consequently, there is a
need for an accessible, theoretically founded formulation
of fundamental aspects of epitaxy, particularly to simplify
prediction of possible epitaxial configurations in these real
and, hence, complex systems.
The development of theoretical descriptions of epitaxy
began with Frank and van der Merwe,[1] who accounted
for the periodic nature of the crystals of differing lattice
parameters and introduced the concept of pseudomorphic
strain, misfit accommodation, and misfit dislocations in one-
dimensional systems. Since then, many models have been
developed from a range of perspectives, usually with models
applicable to interfaces with simple geometry: polyatomic
crystals have rarely been treated.
Generalization of existing theories to complex symmetries
and unit-cell structures has been hampered by the absence
of a framework within which the models and predictions
can be expressed consistently. There is a need for a descrip-
tion that is general, insensitive to anisotropy of elastic prop-
erties, does not rely on simple symmetries, is applicable to
polyatomic systems of any size with least complication, and
whose results and predictions can be visualized in terms of
any preferred or current framework. The reciprocal-space
description given here is a strong candidate. While several
fundamental aspects of this formalism have been applied by
earlier authors, these are significantly extended here.
II. THE MODEL
MAX W.H. BRAUN, Director, is with the Centre for Science Education
and the Department of Physics, University of Pretoria, Pretoria 0002, South
Africa. JAN H. VAN DER MERWE, Professor, is with the Department of
Physics, University of South Africa, Pretoria 0002, South Africa.
This article is based on a presentation in the symposium “Interfacial
Dislocations: Symposium in Honor of J.H. van der Merwe on the 50th
Anniversary of His Discovery,” as part of the 2000 TMS Fall Meeting,
October 11–12, 2000, in St. Louis, Missouri, sponsored under the auspices
of ASM International, Materials Science Critical Technology Sector,
Structures.
Several models yield expressions which approximate the
energy of an epitaxial bicrystal and have been used to predict
epitaxial orientations or interfacial structure by minimizing
the interaction energy (such as in References 1, 3, 5, and 6).
In the geometric limit, both components of the bicrystal
are treated as rigid, retain their bulk or homogeneously
deformed lattices, and are in contact at a single interfacial
plane. On either side of the interface, each component crystal
METALLURGICAL AND MATERIALS TRANSACTIONS A
VOLUME 33A, AUGUST 2002—2485

presents a crystal plane, with a set of translational and rota-
tional symmetries and two-dimensional periodicities. The
set of wave vectors that together form the surface-reciprocal
lattice for each crystal face conveniently describes these
periodicities. It is useful to refer to the component crystals
as substrate and overgrowth, respectively, in particular when
one crystal is extremely thick, whereas the other is only an
atomic layer or so in thickness.
Fletcher[5] has calculated the energy for a bicrystal system
by summing over all the pairwise interactions in the bicrystal
while assuming central potentials such as Morse or Lennard–
Jones. Novaco and McTague[7] developed a linear-response
lattice dynamic formulation based on Fletcher’s model,
which was applied to this rigid limit. Van der Merwe[3]
considered an orientation-dependent contribution to the
energy in a rigid overgrowth and substrate system by using
a truncated Fourier series to express the overgrowth atom-
substrate crystal interaction potential. We present here a
short development of an expression for such a bicrystal
energy, which is essentially a generalization of Van der
Merwe’s approach.[2,3] The central result obtainable from
these approaches is a necessary criterion for the occurrence
of an epitaxial orientation expressed in reciprocal space. It
is an energetically founded geometric criterion imposed by
the crystalline nature of epitaxial systems and is independent
of a particular atomic-force model.
Fig. 1—Transformation parameters for bcc {110} overgrowth (small cir-
cles) on fcc {111} substrate (larger circles).
III. THE EPITAXIAL CRITERION
We extend the relation [1] to deal with an island character-
ized by primitive interfacial unit cells of (2M ϩ 1) ϫ (2N ϩ
1) overgrowth lattice points (adatoms C) on a substrate (S)
with primitive unit cells (atoms A). This can easily be gener-
alized to crystalline-island overlayers involving this and
more complicated cases, as is done subsequently. In this
simple case, the overgrowth lattice sites are arranged as
2M ϩ 1 rows of 2N ϩ 1 points each and are generated by
displacing a single point by all the vectors in the set
The substrate surface may be described in terms of a two-
dimensional Bravais lattice formed by the basis vectors a1
and a2, with lengths of a1 and a2, respectively, and an angle
*
of ␣ between them. An associated reciprocal set a1 and
*
*
a2 is uniquely defined by the condition ai ϫ aj ϭ 2␲␦ij ,
where ␦ij is the Kronecker delta, and by the requirement that
these vectors are all coplanar. The lengths a1 and a2 may be
expressed in terms of the bulk nearest-neighbor distance ann
as ai ϭ Cai ann (i ϭ 1, 2) so defining the scaling factors Ca1
and Ca2.
r
mn ϭ mb1 ϩ nb2: m ϭ ϪM, ϪM ϩ 1, … Ϫ1, 0, 1, … M,
[3]
n ϭ ϪN, ϪN ϩ 1, … Ϫ1, 0, 1, … N
The interaction energy between individual interfacial
overgrowth atoms (adatoms) and the substrate surface is
assumed to be periodically dependent on the adatom position
(r) and is conveniently expressed as a Fourier series:
ϱ
The vectors b1 and b2, with lengths of bi ϭ Cbi bnn (i ϭ
1, 2) and an angle of ␤ between them, form the set of basis
vectors of the Bravais lattice of the overgrowth surface; the
*
*
coplanar vectors, b1 and b2 , define the overgrowth surface-
V(x,y) ϭ
Vqeiqиr
ϭ
Vhkei2␲(hxϩky)
[1]
*
reciprocal lattice with bi ϫ bj ϭ 2␲␦ij . The relationship
͚
͚
{q}
h,kϭϪϱ
between the substrate and overgrowth unit cells, for an exam-
ple system of a bcc(110) overgrowth on an fcc(111) sub-
strate, are shown in Figure 1.
The total interaction energy for the island is obtained by
summing the adatom interaction energy over all the atoms
(hence, all m and n terms) in the overgrowth island, which
yields a geometric series and, after some simplification, is
expressed as
with
*
r ϭ xa1 ϩ ya2,
a2 ϫ n
qhk ϭ ha1* ϩ ka2
n ϫ a1
*
*
a2 ϭ 2␲
a1 ϭ 2␲
,
[2]
Ȋa1 ϫ a2Ȋ
Ȋa1 ϫ a2Ȋ
a1 ϫ a2
with unit normal,
n ϭ
Ȋa1 ϫ a2Ȋ
sin ␲(2M ϩ 1)p sin ␲(2N ϩ 1)q
V ϭ
Vhk
ϫ
[4]
͚
where qhk is a lattice-translation vector of the substrate sur-
face-reciprocal lattice, h and k are integers, and r is a position
vector expressed in terms of the substrate. The Fourier coeffi-
cients can be determined from extensive detailed first-princi-
ple calculations, or by using empirical or semiempirical
atomic potentials.[8,9] Many-body calculations can lead to
values for these coefficients. These single atom–substrate
calculations are of interest in absorption and atom-surface
migration. Calculations have more recently been extended
to atoms belonging to an epilayer using many-body
interactions.[10]
΂
΃
sin ␲p
sin ␲q
h,k
hk
where p and q are functions of h and k, given by the
expressions
p(h, k) ϭ hr11c ϩ kr12s␪ , q(h, k) ϭ hr21s␪␤ ϩ kr22c
␪
␪␤
[5a]
Here, ␪ is the angle between b1 and a1 and determines the
relative orientations of the overgrowth and substrate unit
cells. The ratios
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METALLURGICAL AND MATERIALS TRANSACTIONS A

bi Cbi
ϭ
bnn
ann
IV. GEOMETRIC CONSEQUENCES OF THE
EPITAXIAL CRITERION
rij ϭ
r,
r ϭ
(i, j ϭ 1, 2)
[5b]
aj Caj
A. Matching of Lattice Rows
define the dimensional relationship between the overgrowth
and substrate lattices in terms of the parameter r, the ratio
of nearest-neighbor distances in the respective lattices, which
identifies most readily with the atomic-size ratio introduced
by Bruce and Jaeger.[11] Other transformation parameters in
Eq. [5a] are
Discussion of the ideal epitaxial condition and its geomet-
ric criterion is simplified with a system of indices for direc-
tions and lattice rows in the surface lattice, consistent with
Miller indices in crystallography. These indices may be asso-
ciated with directions and spacings of families of rows of
lattice points. Just as reciprocal lattice vectors of a three-
dimensional lattice act as wave-propagation vectors of plane-
wave fronts which coincide with lattice planes, so in two
dimensions reciprocal lattice vectors propagate line fronts
that coincide with rows of lattice points. The spacing of these
lattice rows is given by the wavelengths of the propagation
vectors, while the components of the reciprocal lattice vector
provide Miller indices for the lattice rows. In particular, the
reciprocal lattice vector qhk ϭ ha1* ϩ ka2* (denoted here
by [h k]*) defines a family of direct lattice rows that may
be indexed as (h k), and the vector qhk is perpendicular to
these rows. The reciprocals of the indices give the intercepts
of the first lattice row of the family along the a1 and a2
vectors, and integer multiples give the intercepts of the oth-
ers. A specific direction in direct space parallel to xa1 ϩ
ya2 is written as [x y]. The spacing of the rows propagated
by [h k]* is given by
sin (␣ Ϫ ␪)
sin ␣
sin ␪
sin ␣
c ϭ
,
s ϭ
␪
␪
[5c]
sin (␣ Ϫ ␤ Ϫ ␪)
sin ␣
sin (␤ ϩ ␪)
sin ␣
s␪␤ ϭ
,
c
␪␤ ϭ
Implicit to the derivation of Eq. [4] is the assumption that
atoms in the overlayer interact independently with the sub-
strate, so that the total may be obtained by simple summation.
An equally important assumption is that the atomic interac-
tion across the interface is of rather short range and involves
the atoms adjacent to the interface only.
Direct calculation of the transformation between the b1
and b2 and a1 and a2 systems (by expressing these vectors
in terms of a common Cartesian coordinate system, and
solving for p and q) shows that p(h, k) and q(h, k) in Eqs.
[4] and [5] are the coordinates of the substrate wave vector
qhk ϭ ha1* ϩ ka2*, expressed in the overgrowth reciprocal
lattice as qhk ϭ p(h, k) b1* ϩ q(h, k) b2*. The energy in
Eq. [4] peaks sharply when p and q are integers. When this
necessary condition is met, the interfacial energy is sharply
minimized. Consequently, the translation vector of the sub-
strate reciprocal lattice qhk must coincide with a translation
vector of the overgrowth reciprocal lattice, qpq. The condition
2␲
ȊqhkȊ
␭hk ϭ
[7]
ϭ 2␲ Ϭ h2a* и a* ϩ 2hka* ϫ a* ϩ k2a* ϫ a*
Ί
1
1
1
2
2
2
In analogy to the zone law of crystallography, it is always
true that
ha1* ϩ ka* ϭ [h k]* Ќ [k h] ϵ ka1 Ϫ ha2
[8]
2
where the overbar indicates a negative index. Lattice rows,
indexed as (h k), lie parallel to the direct lattice direction
[k h].
Consequences of the coincidence of a pair of overgrowth
and substrate reciprocal lattice vectors include the following.
pq
ahk ϭ qpa, i.e., qhk ϵ ha1 ϩ ka2 ϭ pb* ϩ qb2 ϵ q
*
*
*
1
[6]
is a necessary condition for an ideal epitaxial configuration,
which is defined as the orientation and associated lattice
parameters at which the interfacial misfit energy is mini-
mized for a rigid system, as introduced by van der Merwe.[3]
Note that the essence of the rigidity condition is that the
overlayer atomic positions in Eq. [3] form a regular grid.
This also includes a homogeneously deformed lattice (epi-
layer). At equilibrium, systems with ideal lattice parameters
will be found in the orientation yielding the least interfacial
energy. This necessary condition for interfacial-energy mini-
mization has previously been obtained from a different
energy model by Fletcher.[5]
(1) If the wave vectors are parallel, then so are the lattice
rows they propagate. When crystals are aligned in ideal
epitaxial orientation, lattice rows of substrate and over-
growth are aligned in a parallel orientation.
(2) The aligned lattice rows have the same spacing in a
direction perpendicular to the lattice rows. Since qhk
ϭ
pq
qpq, it follows that ␭ ϭ ␭ where
hk
2␲
ȊqpqȊ
pq
␭
ϭ
[9]
ϭ 2␲ Ϭ p2b* ϫ b* ϩ 2pqb* ϫ b* ϩ q2b* и b*
Ί
1
1
1
2
2
2
As discussed subsequently, the condition [6] is equivalent
to a criterion that epitaxy occurs when sets of lattice rows
in the overgrowth and substrate are parallel and have equal
spacing in the interface. Atomic rows, as opposed to lattice
rows, are governed by the reciprocal-lattice criterion if the
reciprocal lattice is appropriately modified by the inclusion
of structure factors to account for nonprimitive unit cells
and translations. Only reciprocal lattice translation vectors,
which do not cause the combined structure factors to vanish,
are relevant.
This is analogous to the Bragg condition of crystal-
lography.
(3) The spacing of atoms along the rows is not matched by
this single criterion. Matching a second spacing also
produces full two-dimensional coherency.
B. Construction
The matching criterion of Eq. [6] is analogous to the
von Laue condition of diffraction theory[12] and leads to a
METALLURGICAL AND MATERIALS TRANSACTIONS A
VOLUME 33A, AUGUST 2002—2487

through which the overgrowth needs to rotate in relation to
the substrate to achieve epitaxy. This angle, ␪R , is given by
qhk ϫ qpq
ȊqhkȊȊqpqȊ
cos (␪R) ϭ
[10]
The vector cross product qpq ϫ qhk defines the sense of the
rotation. The nearest-neighbor ratio that allows a particular
pair of vectors to match in length can be calculated from
the condition ␭ ϭ ␭pq, when the parameters Cb1 and Cb2
hk
and the unit-cell angle ␤ are unchanged.
The pair of values ␪ and r (Eq. [5b]) associated with a
R
given pair of surface structures, together with a choice of
the orientation for which ␪ ϭ 0, allows ideal epitaxial config-
urations to be specified uniquely for a given pair of interfa-
cial planes. Such a convention was used by Bruce and
Jaeger[11] and van der Merwe.[3]
To express the epitaxial configuration as parallel crystallo-
graphic directions, the lattice row directions must be written
in terms of the (three-dimensional) crystallographic indices
of the surface unit-cell vectors: if the unit-cell vectors are
given as a1 ϭ [u v w] and a2 ϭ [U V W];, respectively, the
reciprocal lattice vector [h k]* (itself perpendicular to [k h])
relates to the row of atoms which lies parallel to the direction
ka1 Ϫ ha2 ϭ [ku-hU kv-hV kw-hW]. Similarly, the matched
overgrowth reciprocal lattice vector provides the crystallo-
graphic indices of the matched direction and the index of
the row of lattice points whose spacing is matched with the
substrate, but along which lattice points are not necessar-
ily matched.
As an example, consider the bcc(110) surface and its
epitaxy with an fcc(111) surface, as in Figure 1. Surface
unit-cell basis vectors are a1 ϭ 1/2[1 1 0] and a2 ϭ 1/2[0 1 1]
for the fcc substrate and b1 ϭ [1 1 0] and b2 ϭ [0 0 1] for
the bcc overgrowth, respectively. For perfect matching in
the well-known Nishiyama–Wassermann orientation, the
overgrowth reciprocal lattice vector [2 0]* coincides with
the substrate reciprocal lattice vector [1 0]*. In surface coor-
dinates, the lattice row, whose spacing is matched is then
perpendicular to the respective surface reciprocal lattice vec-
tors, is [0 2] and [0 1] for the overgrowth and substrate,
respectively. Hence, the crystallographic directions of the
overgrowth and substrate, which are parallel, are [0 0 1] and
[0 1 1], respectively. Lattice positions along these directions
are not matched, but the spacing of rows in the perpendicular
(overgrowth 1/2[1 1 0] and substrate 1/4[2 1 1]) directions
are equal.
Fig. 2—The “Ewald circle” corresponding to the substrate wave vector
[1 0]*. Both overgrowth reciprocal lattice vectors [2 0]* and [1 1]* lie
close to the circle. Note also that although the unit cell of the bcc(110)
surface was rectangular, the effect of the central point has been to remove
some reciprocal lattice points, illustrating the effect of the structure factor.
geometrical realization similar to the Ewald construction,
as illustrated in Figure 2. This simplifies the qualitative
discussions of epitaxy of real crystals and simultaneously
provides a visualization and a tool for quantitative predic-
tions of epitaxial orientations.
The steps in the construction may be described as follows.
(1) Choose the substrate lattice row which is to be matched
with the overgrowth. Express its direction as the sub-
strate direct lattice direction [k h]. Plot qhk ϭ [h k]* to a
suitable scale in the substrate reciprocal lattice. Substrate
structure factors, to be considered subsequently, may,
depending on their sign and magnitude, respectively,
nullify the given vector qhk , or reduce the epitaxial
potency associated with it. This must be taken into
account.
(2) Draw a circle, centered at the origin, through the end
of qhk . This circle forms the locus of the endpoint of
qhk when this vector is rotated through 360 deg.
(3) Plot the overgrowth reciprocal lattice to the same scale
and in any suitable orientation, but with the origin at
the start of the vector qhk . (Again, take account of struc-
ture factors.)
V. MISFIT AND MISFIT ACCOMMODATION
A. Measures of Misfit
Misfit between two crystals with identical structure, meet-
ing at the same crystallographic plane but differing only in
an overall scale factor, may intuitively be expressed by a
single ratio of lattice parameters. This simple description
breaks down when surface symmetries and orientations dif-
fer. The row-matching criterion (Eq. [6]) provides a general
framework for defining misfit of general interfaces.
Two classes of misfit can be identified. These are orienta-
tional misfit and dimensional misfit (general misfit contains
aspects of both). The orientational misfit is given by the
angle between nearly coinciding reciprocal lattice vectors
as ␪R , defined by Eq. [10].
Any overgrowth reciprocal lattice point qpq that lies on
this circle will describe an overgrowth reciprocal lattice
translation vector, equal in length to qhk , and, hence, yields
the p and q values which satisfy the epitaxial criterion of
Eq. [6]. In addition, when the overgrowth lattice is rotated
so that a lattice point qpq lying on the circle coincides with
qhk , the overgrowth and substrate atomic rows with the same
spacings in both lattices that are perpendicular to qpq and
qhk will be parallel. The overgrowth and substrate will be
in an ideal epitaxial orientation.
The angle between the vectors qpq and qhk gives the angle
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METALLURGICAL AND MATERIALS TRANSACTIONS A

Measures for the dimensional misfit have been introduced
by several authors and are convenient under specific condi-
tions. Consider the following three expressions for misfit,
where ␭ and ␭ are the substrate and overgrowth lattice
while predictions of strains in perpendicular (free) directions
and associated energy densities may be calculated within
the assumptions of elasticity theory.[2,17,18] This requires esti-
mates of elastic constants for the overgrowth region, and
even if they are somewhat uncertain for very thin over-
growths, the energy density does provide a measure of the
likelihood that a particular configuration will be observed:
the greater the deformation energy, the less the likelihood
that the configuration will be observed. Elastic constants
must be transformed to a coordinate system suitable for the
overgrowth as a two-dimensional system, and plane-stress
boundary conditions must be enforced to account for the
free surface parallel to the interfacial plane to yield a suitable
expression for the energy density in terms of two-dimen-
sional strains:
a
b
spacings, respectively.
␭ Ϫ ␭
␭a Ϫ ␭
␭ Ϫ ␭
a
b
b
a
b
fa ϭ
;
fb ϭ
;
fab ϭ
␭
a
␭
b
1
2
(␭ ϩ ␭b)
a
[11]
The first was introduced by Frank and van der Merwe[1]
and applies when the substrate is considerably thicker than
the overgrowth and is approximately rigid while the misfit
is referred to the substrate lattice spacing, ␭ ϭ ␭ (Eq.
a
hk
pq
[7]), while ␭ ϭ ␭ (Eq. [9]) is the overgrowth spacing.
b
The second was used by Matthews[13] and applies most con-
veniently when the overgrowth strains to partially match the
substrate, and the misfit calculated by this expression has
the same sense as strain when related to the initial overgrowth
lattice spacing. The third averages the lattice spacings and
was used by van der Merwe[14] and Jesser and Kuhlmann-
Wilsdorf[15] to describe systems in which overgrowth and
substrate both strain. While the first and third definitions
are actually the negatives of those of the original authors,
they are preferred here for the convenience of having the
same sense for misfit and strain. Besides the functional
advantages of particular definitions of dimensional misfit,
each expression is related to a particular choice of Burgers
vector. The Burgers vector expresses directly the closure
failure (per misfit dislocation) associated with a dislocation
array that may accommodate the misfit. These aspects are
discussed subsequently.
1
2
␰ ϭ (D11␧2 ϩ D22␧2 ϩ D33␥2xy) ϩ D12␧ ␧ ϩ D13␧x␥xy
x
y
x y
[12]
ϩ D23␧y␥xy
where the D terms are appropriately transformed, two-
dimensional elastic constants. It is sufficient to point out
here that, where the intention is to calculate the spacing,
orientation, and character of arrays of misfit dislocations
which accommodate residual strain, the strained lattice
parameters can be substituted into expressions [1] through
[4].
C. Misfit Dislocations and Misfit Vernier
The third misfit accommodation mode, besides simple
rotation and homogeneous strain, occurs when the energy
associated with misfit strain exceeds that to be gained by
exact matching. Local relaxation will occur if possible, but
overall lattice periodicities will not be the same in over-
growth, and substrate and reciprocal wave vectors will not
coincide. Local relaxation does imply alternate regions of
good and poor matching. This pattern is associated with
misfit dislocations if the regions of fit are larger than those
of the mismatch. “Misfit Vernier” is the term used by van
der Merwe[3] to characterize the pattern that occurs if the
regions of good and bad fit are of nearly equal size with
essentially no relaxation. In both cases, these patterns repeat,
and the period is known as the dislocation spacing or vernier
period, as appropriate. The period can be calculated by
exploiting the analogy between misfit dislocation arrays aris-
ing from the growth of crystals with different lattice spacings
and the superposition of two waves of different wavelengths
to produce beats. Similar geometric considerations have
shown these associations with Moire´ patterns both in direct
space, as observed by Bollmann,[16] and in reciprocal space,
as observed by Jesser.[19]
B. Misfit Accommodation by Homogeneous Strain in
One and Two Dimensions
If rows of atoms nearly match in orientation and spacing
(the corresponding reciprocal lattice vectors nearly coin-
cide), the overgrowth (with the substrate treated as rigid)
may be expected to deform to reduce mismatch if the energy
cost due to strain is less than the energy gained when the
register is improved.
If the shape of the minimum in the energy function (Eq.
[4]) is a very narrow well, then the overgrowth effectively
distorts until exact matching is achieved, while for a wide
well, the energy will be minimized without exact matching.
The overgrowth structure deforms homogeneously toward
adopting the substrate lattice spacing. The term pseudomor-
phism has been adopted if matching has been achieved, and
the strain associated with the deformation has been termed
misfit strain.
Matching may involve a single direction in the crystal in
which the overgrowth strains to match the spacing of a
family of rows, while the lattice spacing along the rows
remains mismatched (one-dimensional pseudomorphism).
Alternatively, matching may be achieved in all directions
(two-dimensional pseudomorphism). This may include coin-
cidence matching in the sense of Frank or O-lattice matching
in the sense of Bollmann,[16] where some lattice rows may
be skipped in the overgrowth or the substrate.
From the superimposed wave analogy, the beat wave vec-
tor is given by the difference of the wave vectors (Figure 3):
␦q ϭ qb Ϫ qa
[13]
where qb and qa are nearby wave vectors from the over-
growth and substrate, respectively. The wave vector ␦q
determines the spacing of the beats, or dislocations, and
their orientation. The spacing (similar to Eq. [7]) is
The strains necessary to achieve matching may be calcu-
lated directly from the condition for ideal epitaxy (Eq. [6]),
METALLURGICAL AND MATERIALS TRANSACTIONS A
VOLUME 33A, AUGUST 2002—2489

Referring to Figure 3, when ȊqaȊ ϭ ȊqbȊ, there is no dimen-
sional difference between the lattices, but only an orienta-
tional difference. Here, qr Ќ ␦q, and a pure screw dislocation
array is described, with the Burgers vector parallel to the
dislocation line, in correspondence with the properties of a
general dislocation and misfit dislocation theory.[14,15,19] The
other extreme occurs when qb ሻ qa with no orientational
difference, but only dimensional misfit. This produces a pure
edge misfit dislocation with the Burgers vector perpendicular
to the dislocation line. Characterization of the screw and
edge character of the misfit dislocations is a property of
the Burgers vector Br , based on the reference lattice only,
together with the sense of the dislocation array, Z, given by
the difference vector.[20]
A convenient approach may be to describe dislocation
orientations and Burgers vectors crystallographically in
whichever system (overgrowth or substrate) is preferred, but
to determine the analytically important properties, such as
the fraction of edge or screw character, from the reference
lattice. Table I summarizes several of the important
parameters.
Fig. 3—(a) through (c) Characteristics of misfit dislocations, which can
be determined from the reciprocal lattice. Note how the reference Burgers
vector Br and the dislocation line sense vector Z determine the character
of the misfit dislocation array uniquely. These are obtained from the vectors
qr and ␦q, where Br ሻ qr , while Z Ќ ␦q.
2␲
Ȋ␦qȊ
2␲
2␲
(q Ϫ q ) ϫ (q Ϫ q )
Ί
Variations which can be considered include dislocation
interactions whose Burgers vectors of arrays with equal spac-
ing can be added to produce other, equivalent arrays or
distended rows of misfit.
␭D ϭ
ϭ
ϭ
␦q ϫ ␦q
Ί
a
b
a
b[14]
␭ ␭
a
b
ϭ
2
2
␭ ϩ ␭ Ϫ 2␭ ␭ cos ␪
Ί
a
b
a
b
R
while dislocation lines are parallel to the wave front propa-
gated by the wave vectors. Applying Eq. [8], the lines of
coinciding phases (the wave fronts) lie along direct-space
lattice directions given by Z ϭ [␦q2, ␦q1], which gives the
line sense of the dislocation and may be expressed in sub-
strate or overgrowth coordinates. The dislocation lines them-
selves intersect the basis vectors at integer multiples
VI. STRUCTURE FACTORS
The foregoing considerations on ideal epitaxial configura-
tions and their reciprocal lattice description and construction
are generalized to include island displacements and nonprim-
itive overlayer and substrate surface unit cells.
1/(␦q)a a1 and 1/(␦q)a a2 in the substrate coordinates, where
A. Substrate Structure Factors of Nonprimitive (Atomic
Sites A and B) Substrate Surface Unit Cells
1
the substrate reciproc2al lattice coordinates of the vector ␦q
are used. Similarly, overgrowth coordinates provide the
intercepts in the overgrowth lattice.
Consider a primitive overgrowth island, formed of C
atoms in position r, while the substrate has atomic sites (or
atoms) of two kinds, A at the origin and B in position r+,
as measured in the surface plane. The overgrowth atom C
is in a position of r Ϫ r+ with respect to B. The interaction
energy of C with respect to the pairs A and B is accordingly
given by (V ϵ VC ), from Eq. [1]:
Misfit dislocations and misfit vernier share the property
that the superimposed lattices are in exact phases repeatedly
after a constant distance of ␭D ϭ (1/␭ Ϫ 1/␭a)Ϫ1 ϭ P␭r ϭ
b
(P ϩ 1/2)␭ ϭ (P Ϫ 1/2)␭ (P is an integer when the ratio
b
a
between ␭ and ␭ is rational) from which it follows that
a
b
1/␭ ϭ 1/2(1/␭ ϩ 1/␭b) and that the misfit fab ϭ 1/P. This
r
a
Vϩhk
ϱ
is generalized to two dimensions by introducing a reference
lattice (or, continuing the beat analogy, the wavelength of
the superimposed wave) as
)ϩk(yϪy ))
ϩ
ϩ
V(x, y) ϭ
Vhk ei2␲(hxϩky)
ϩ
ei2␲(h(xϪx
͚
΂
΃
Vhk
h,kϭϪϱ
[17]
ϱ
ϵ
VhkFhkei2␲(hxϩky)
1
2
͚
qr ϭ (qb ϩ qa)
[15]
h,kϭϪϱ
One can introduce the following definitions:
The Burgers vector can be expressed in terms of the
reference lattice qr or in terms of qa or qb as
Vϩhk
Ϫi2␲(hx ϩky
)
ϩ
ϩ
Fhk ϭ 1 ϩ ␻ e
,
␻ ϵ
hk
hk
Vhk
qi
i ȊqiȊ
Fhk ϭ Fhck Ϫ iFhsk ϭ 1 ϩ ␻ cos 2␲(hkϩ ϩ kyϩ)
[18]
Bi ϭ ␭
,
with i ϭ a, b, or r
[16]
hk
Ϫ i␻ sin 2␲(hxϩ ϩ kyϩ)
hk
Definitions of the Burgers vector in terms of overgrowth or
substrate lattice spacings have the convenient feature that
crystallographic directions may be associated with them.
The reference lattice definition has the useful characteristic
that it expresses the character of the misfit dislocation
directly and that the symmetrical measure of dimensional
misfit, fab , is directly related to it.
to derive compact expressions for the potential given in Eq.
[17] as well as real and imaginary parts which correspond
to interactions described by cosine and sine functions,
respectively. (The interaction potentials for the cosine and
sine series are assumed to have real coefficients in the expres-
sions given in Eq. [19].)
2490—VOLUME 33A, AUGUST 2002
METALLURGICAL AND MATERIALS TRANSACTIONS A

Table I. Summary of Misfit Dislocation Descriptors Obtainable from the Reciprocal Lattice
Absolute Misfit
Dislocation Spacing
Reference Lattice
␭ ␭
2␭ ␭
a b
2␲
Ȋ␦qȊ
1
2
a
b
␦q ϭ qb Ϫ qa ϭ [␦q1, ␦q2]*
␭D ϭ
ϭ
qr ϭ (qb ϩ qa) with ␭r ϭ
␭ ϩ ␭2 Ϫ 2␭ ␭ cos ␪
␭ ϩ ␭2 ϩ 2␭ ␭ cos ␪
2
2
Ί
Ί
a
b
a
b
R
a b a b R
Line Sense Vector
Burgers Vectors
Character
qi
i ȊqiȊ
Z ϭ [␦q2, ␦q1]
Pure screw: qr Ќ ␦q
Pure edge: q ሻ ␦q
Bi ϭ ␭
, with i ϭ a, b, or r
Screw fraction ϭ cos2␥, where ␥ is the angle between Br
and the line sense vector, Z
ϱ
lattice, the mth and nth lattice points in the overgrowth are
given as mb1 ϩ nb2 ϩ x0a1 ϩ y0a2. It is seen that the
translation term is common to all overgrowth lattice points,
and, correspondingly, acts as a common multiplier of the
quotient functions in Eq. [17] and produces a further set of
structure factors:
Vc(x, y) ϭ
Vhk(Fhck cos 2␲(hx ϩ ky)
Vhk(Fhck sin 2␲(hx ϩ ky)
͚
h,kϭϪϱ
ϩ Fshk sin 2␲(hx ϩ ky))
ϱ
[19]
Vs(x, y) ϭ
͚
h,kϭϪϱ
o
co
so
Fhk ϭ ei2␲(hx ϩky ) ϭ Fhk ϩ iFhk
0
0
Ϫ Fshk cos 2␲(hx ϩ ky))
co
F
F
ϭ cos 2␲(hxo ϩ kyo)
ϭ sin 2␲(hxo ϩ kyo)
[22]
hk
The extension to more than one additional feature is straight-
forward and has more terms in each of the structure factors
defined in Eq. [18], without changing the essential form of
Eqs. [17] and [19].
so
hk
Expression [21] now becomes
Ј
Ј
V ϭ
VhkFhkFohk ei␲(M ϪM)p ϫ ei␲(N ϪN)q
͚
΂
h,k
B. Overgrowth Structure Factors of Island Size
[23]
sin␲(M ϩ MЈ ϩ 1)p sin␲(N ϩ NЈ ϩ 1)q
In this section, the inclusion of structure factors due to
the overgrowth island is developed. As a first step, consider
an asymmetrical island which is constructed from atoms at
overgrowth lattice points defined by all the vectors in the set:
ϫ
ϫ
΃
sin␲p
sin␲q
hk
D. Nonprimitive Overgrowth Unit Cells
r
mn ϭ mb1 ϩ nb2:
Nonprimitive unit cells are characterized by the presence
of more than one atom in a unit cell, or more than one lattice
point in a unit cell. These are at positions displaced from
overgrowth lattice positions. We derive the effect of this
displacement by considering an atom displaced from a lattice
site by x+b1 ϩ y+b2. If the total interaction energy of these
displaced atoms is considered, every m term is shifted by
x+ and every n term by y+. This appears again in every
exponent and can be taken out as common structure factor.
In the simplest case, the essential symmetry of the substrate
potential is retained, but may be subject to a multiplier, such
as ␬. Atoms at lattice positions do not contain this term,
and a two-atom unit cell thus produces an unaltered energy
expression, plus a second expression in which each Fourier
term is multiplied by ␬ and a displacement factor. As the
displacement is expressed in terms of the basis vectors of
the overgrowth, the displacement term is most compactly
expressed in terms of the pair p, q. Overgrowth structure
factors may then be defined as
m ϭ ϪM, ϪM ϩ 1, … Ϫ1, 0, 1, … MЈ Ϫ 1, MЈ, [20]
n ϭ ϪN, ϪN ϩ 1, … Ϫ1, 0, 1, … NЈ Ϫ 1, NЈ
(Here MЈ, M, NЈ, and N are not necessarily equal.)
Summing individual interaction terms of the exponential
form given by Eq. [1] leads directly to
Ј
Ј
V ϭ
VhkFhk ei␲(M ϪM) ϫ ei␲(N ϪN)q
͚
΂
h,k
[21]
sin ␲(M ϩ MЈ ϩ 1)p sin ␲(N ϩ NЈ ϩ 1)q
ϫ
ϫ
΃
sin␲p
sin␲q
hk
The leading exponential expressions become 1 for a symmet-
rical lattice with M ϭ MЈ and N ϭ NЈ. The quotients in
the sine functions, as well as the leading exponentials, are
responsible for island-size effects in the misfit energies. The
delta function–like behavior of the quotients again leads
directly to the ideal epitaxial configurations, as obtained
earlier.
ϩ
ϩ
ϩky
Fpq ϭ 1 ϩ Fϩpq ϭ 1 ϩ ␬ei2␲(px
) ϭ Fpcq ϩ iFpsq
pq
ϩ
ϩ
Fpcq ϭ 1 ϩ Fc ϭ 1 ϩ ␬ cos 2␲(px ϩ qy )
[24]
ϩ
pq
ϩ
ϩ
Fpsq ϭ Fs ϭ ␬ sin 2␲(px ϩ qy )
ϩ
C. Structure Factor Expressing Translation of the
Overgrowth Island
These structure factors are readily generalized to a basis
of more than one atom by including more terms in each
expression.
If the origin of the overgrowth island undergoes a displace-
ment of x0a1 ϩ y0a2 from the origin of the substrate surface
METALLURGICAL AND MATERIALS TRANSACTIONS A
VOLUME 33A, AUGUST 2002—2491

E. Most General Structure Factors
candidates for epitaxial configurations for particular pairs
of overgrowth and substrate.
The general expressions for the misfit energy of a multisite
substrate, displaced, asymmetric, multiatom overgrowth is,
finally, given by
These qualitative considerations can provide a preliminary
ordering of candidate ideal epitaxial orientations into a hier-
archy.[21] The ability to predict interfacial character can be
improved somewhat if some broad quantitative considera-
tions are added into the calculations. For example, elastic-
energy densities (Eq. [12]) for the one- and two-dimensional
pseudomorphic fit modes arising from the predicted ideal
epitaxial configurations can be calculated; misfit dislocation
strain energies can be estimated from well-known expres-
sions, such as those of van der Merwe[1,15] and Matthews;[13]
and structure factors (including size effects) can be
precalculated.
Ј
Ј
V ϭ VhkFhkFohkFpqei␲(M ϪM)p ϫ ei␲(N ϪN)q
͚
h,k
[25]
sin ␲(M ϩ MЈ ϩ 1)p sin ␲(N ϩ NЈ ϩ 1)q
ϫ
ϫ
΂
΃
sin ␲p
sin ␲q
hk
where the real part of the expression refers to interactions
defined in terms of cosines only, and the imaginary part
refers to interactions in terms of sines only (Vc and Vs of
Eq. [19] respectively).
The total interfacial energy given by (Eq. [5]) shows
strong minima at ideal epitaxial configurations. This energy
reduction provides the driving force to both orient and
deform the unit cells on either side of the interface. While
wave vectors match in reciprocal space and produce a peri-
odic structure in direct space, matching of a particular pair
of wave vectors also produces matching with multiples of
these vectors—there is also a periodic structure produced
in reciprocal space. The misfit-energy reduction is produced
not only with the low-order Fourier coefficient, but also to
an infinite number of higher-order terms. The higher the
fraction of all possible vectors that are matched, the greater
the energy reduction. Therefore, the density (in reciprocal
space) of matched vectors is a qualitative parameter for
prediction of good epitaxial candidates.
The combined structure factors may be shown to be
co
so
((FhckF ϩ FshkF )(FcpqFcM,N Ϫ FpsqFMs ,N
)
hk
hk
[26a]
[26b]
[26c]
so
co
Ϫ (FhckF Ϫ FhskF )(FpsqFcM,N ϩ FspqFMs ,N))
hk
hk
for the real part, and
((FhckF Ϫ FshkF )(FcpqFcM,N Ϫ FpsqFMs ,N
)
so
co
hk
hk
co
so
ϩ (FhckF ϩ FhskF )(FpsqFcM,N ϩ FcpqFMs ,N))
hk
hk
for the imaginary part, with
FMc ,N ϭ cos ␲((MЈ Ϫ M)p ϩ (NЈ Ϫ N)q) and
FMs ,N ϭ sin ␲((MЈ Ϫ M)p ϩ (NЈ Ϫ N)q)
The misfit energy reduction is directly proportional to the
Fourier coefficient Vhk . Stoop has shown that these Fourier
coefficients decrease approximately exponentially with
order of the Fourier term. The order has been defined as
ȊhȊ ϩ ȊkȊ by Stoop,[8,22] which is more specifically expressed
by the length (in reciprocal space) of wave vectors. It follows
that the shorter the wave vectors which are matched, the
more likely it is that the epitaxial orientation will be
observed. It is true, though, that the criterion has to be
applied with due caution, as Fourier coefficients of the same
order may differ appreciably in magnitude and may even
differ in sign.[23] The shortest wave vectors provide the long-
est range of order, and, hence, ideal epitaxial configurations
can be ordered by the length of reciprocal wave vectors,
or order.
Pseudomorphism is predicted for many epitaxial pairs and
is a well-known concept since Frank and van der Merwe’s
articles of 1949.[1] The strain energy associated with the
deformation of the overgrowth to matching the substrate
must be obtained from the reduction of the misfit energy
from the mismatched case. The smaller this energy, the
greater the chance of the epitaxial configuration being real-
ized in practice. The elastic-strain-energy densities (Eq.
[12]) can be used to rank possible pseudomorphic configura-
tions: the higher the strain-energy density, the less likely is
the configuration to occur.
The structure factors due to the nonprimitive nature of the
unit cells, namely, Fhk and Fpq, will be zero for many combi-
nations of the integers h,k and p,q and, thus, remove recipro-
cal lattice points from the substrate and overgrowth
reciprocal lattices, respectively. Added atoms reduce the
possible epitaxial-matching configurations otherwise pre-
dicted from consideration of the lattices alone. The transla-
tion structure factor Fohk determines the phase of the
overgrowth with regard to the substrate. A bad choice may
put the overgrowth and substrate into direct opposition rather
than coincidence and is characteristic of a rigid model. (This
is significant when misfit dislocations are not allowed.)
In practice, when h,k and p,q are integers (as happens in
ideal epitaxial configurations), the structure factors can be
precalculated, only the relevant Vhk terms need to be retained
in the interaction potential, and some complexity can be
removed from analytical descriptions of even complex struc-
tural pairs. In all cases, the role of the sine quotients in p
and q, which lead to matching of rows of atoms in ideal
epitaxial configurations, remains critically important.
VII. PREDICTION OF INTERFACIAL
CHARACTER
Exact prediction of the final interfacial structure would
require a calculation that includes details of both the dynam-
ics of growth and detailed knowledge of interatomic many-
body interactions, or, within simplified models, knowledge
of elastic constants of the systems and the active Fourier
coefficients of the interaction potential (Eq. [1]). However,
several qualitative considerations can be applied with the
geometric considerations which are a consequence of the
epitaxial criterion (Eq. [6]) to identify (and rank) strong
Polyatomic systems include atoms that exhibit varying
bonding to the substrate (as expressed through the parameter
␬), or one might include bonding sites of varying strength
in the substrate potential, for example, to model stacking-
fault energies (the parameter ␻). These are included in the
structure factors of Eqs. [18] and [24]. The smaller or more
negative the structure factors (Eq. [26]) the less likely is the
2492—VOLUME 33A, AUGUST 2002
METALLURGICAL AND MATERIALS TRANSACTIONS A

configuration, although it is to be noted that the effects of
translation (Eq. [22]) of the unit cell must also be fully
examined, as appropriate translations may change the sign
of the interaction term.
residual misfits can be determined by identifying and exam-
ining reciprocal lattice points which are in near-match condi-
tions. These provide the parameters of expected misfit
dislocation arrays (Table I), against which the actual interfa-
cial structure can be evaluated.
Finally, size effects must be considered within the context
of the growth mechanism and history. When the islands are
small, two effects on the misfit-energy function are evident.
First, the quotients of sines in Eq. [5] are not sharp, and the
growing island can conceivably orient and strain to match
the substrate with a less sharply defined orientation than
would apply to a large island. Second, the sine quotients
themselves show secondary minima that, although shal-
lower, may orient the growing overgrowth island toward
them.[3] Linked to the size effect is the effect of higher-order
terms in the Fourier expansion (long-wave vectors). The
higher the order of the wave vectors, the greater the number
of candidate configurations (now properly described as coin-
cidence configurations) which are close to one another. The
minima of several of these configurations may indeed over-
lap while islands are small, leading to a range of orientations
within which the island may grow during its early stages.
The bcc(110)/fcc(111) interface provides an example of
effects which may be ascribed to growth history. Two well-
described configurations are close to one another, known
as the Nishiyama–Wassermann and the Kurdjumov–Sachs
configurations. Calculations from reciprocal space show that
if an island is originally in two-dimensional coherence, it is
strained to match the reciprocal pairs associated with both
configurations. However, once the island gets too large to
match both (the strain energy being too high), a rotation of
5.26 deg is needed to retain a match with the Kurdjumov–
Sachs pair, while perfect orientation with the Nishiyama–
Wassermann configuration is achievable without rotation.
Gaigher and van der Berg[24] have reported observations on
substances which fall between the ideal lattice parameters
for the Kurdjumov–Sachs and the Nishiyama–Wassermann
configurations. They show that orientations found lie in a
small range of 2 deg around the former orientation, while
the Nishiyama–Wassermann configuration is usually sharply
defined. This range may be ascribed to growth history if the
small island assumes near-pseudomorphism in early stages,
then ceases to match with one of the vector pairs as the
energy minimum becomes more sharply defined with larger
island size. Any impediment to rotation would mean that
the island is discovered away from an exact orienta-
tion. Also, the Fourier coefficients responsible for the
Kurdjumov–Sachs orientation have larger magnitudes than
the two coefficients—one of which is positive—responsible
for the Nishiyama–Wassermann orientation. Consequently,
the range of misfit within which the coefficients for the
Nishiyama–Wassermann orientation is effective is nar-
rowed, and narrows down faster with increasing thickness,
than that generating the Kurdjumov–Sachs orientation.[25]
The qualitative criteria allow a hierarchy of possible epitaxial
configurations to be established, which allows some predic-
tion of likely candidates.
VIII. CONCLUSIONS
The reciprocal-space condition qpq ϭ qhk , which expresses
row matching as a necessary condition for exact epitaxial
matching, has produced a powerful technique for analyzing
epitaxial systems. Although it expresses a geometric rela-
tionship, it is obtained from energy considerations. When
these vectors do not coincide, their difference can be used
to calculate the orientational and dimensional misfits, the
misfit strain and misfit dislocation structures including
Burgers vectors, the spacings and orientation of dislocation
arrays, and the edge and screw character in any given inter-
face within a single formulation.
General considerations lead to a qualitative hierarchy of
ideal epitaxial configurations which can be ordered by the
density of matched reciprocal lattice points, length of
matched wave vectors, elastic strain energy densities, struc-
ture factors, size effects, and dislocation spacings.
The strength of the formalism lies in its generality, the
uniqueness with which an epitaxial configuration can be
described, and the quantity of detail that can be obtained
from it. The disadvantages are that the two-dimensional
nature requires vector manipulations and that, inherently,
reciprocal space is not quite as intuitively natural as direct
space. Favorable, once more, is that descriptions of periodic
structures generally yield simpler expressions in reciprocal
space, and that is exploited here.
A computer program[26,27] (known as Orpheus since 1985
and not related to the commercial software of the same
name) for analyzing epitaxial interfaces and using all the
relationships given here can be obtained from the authors
(MWHB: mbraun@scientia.up.ac.za). It is written in Micro-
soft Quick-Basic* (MSDOS and MACINTOSH** versions).
*Microsoft Quick-Basic is a trademark of Microsoft Corporation.
**MACINTOSH is a trademark of Apple Computer Corp., Cupertino,
CA.
It automates a search for candidate epitaxial configurations
with the construction of the Ewald type described in this
article, provides strains for one- and two-dimensional coher-
ency, strain energy densities, and values for structure factors,
and gives a full list of crystallographic information on
Burgers vectors, dislocation spacings, and screw and edge
characters that can be expected.
ACKNOWLEDGMENT
This work forms part of a Ph.D. Thesis of one of us
(MWHB) and was finalized while we were visiting at the
Department of Materials Science and Engineering, Univer-
sity of Virginia.
In addition to providing a basis for the prediction of
candidates for epitaxial configurations, the reciprocal space
considerations provide a tool for the analysis of experimental
observations. If, for example, a system is known to have a
particular (strained) lattice parameter, this may be used to
calculate the reciprocal spaces of the overgrowth, and the
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METALLURGICAL AND MATERIALS TRANSACTIONS A
VOLUME 33A, AUGUST 2002—2493

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