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Structural transitions in epitaxial overlayers
R. Bruinsma, A. Zangwill
To cite this version:
R. Bruinsma, A. Zangwill. Structural transitions in epitaxial overlayers. Journal de Physique, 1986, 47 (12), pp.2055-2073. �10.1051/jphys:0198600470120205500�. �jpa-00210400�
Structural transitions in epitaxial overlayers
R. Bruinsma and A. Zangwill (*)
Department of Physics, Solid State Science Center, University of California, Los Angeles, CA 90024, U.S.A.
(*) School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.
(Requ le 27 juin 1986, accept6 le 22 ao4t 1986)
Résumé. 2014 Nous présentons une théorie de la croissance épitaxiale adaptée aux matériaux de surcouche qui peuvent adopter une structure cristallographique métastable volumique avec une constante du réseau assez
différente de celle de la structure stable de l’ état fondamental. Ceci requiert une généralisation de la théorie traditionnelle de Frank et Van der Merwe au régime où la surcouche répond non linéairement à une grande
contrainte de cisaillement extérieure. Des résultats nouveaux sont obtenus pour la transition commensurable- incommensurable (pour des couches minces) et pour la transition cohérente-incohérente (pour des couches
épaisses). Ils sont présentés sous la forme de diagramme de phase de structure en fonction du défaut
d’ajustement épitaxial et de l’épaisseur de la surcouche. Nous prédisons que l’analogue cristallographique de la décomposition spinodale peut exister dans certaines gammes de très grandes valeurs du défaut d’ajustement.
La nature de l’épitaxie métastable stabilisée par le substrat est clarifiée, et une analogie avec les transitions de
phase martensitiques est exploitée pour prédire la microstructure qui doit accompagner la transformation de la surcouche métastable à la structure de l’état fondamental pour des surcouches suffisamment épaisses.
Abstract. 2014 We present a theory of epitaxial growth appropriate to overlayer materials that can adopt a bulk
metastable crystal structure with a lattice constant significantly different from that of the stable ground state
structure, as is common for metal epitaxy. This requires a generalization of the traditional Frank and van der Merwe theory to a regime where the overlayer responds nonlinearly to the large shear stress exerted by the
substrate. New results are obtained for the commensurate-incommensurate transition (for thin layers) and the
coherent-incoherent transition (for thick layers) that are presented in the form of a structural phase diagram as
a function of epitaxial misfit and overlayer thickness. We predict that a crystallographic analog to spinodal decomposition can occur in certain ranges of large misfit. The nature of substrate-stabilized metastable
epitaxy is clarified and a further analogy to martensitic phase transitions is exploited to predict the
microstructure that must accompany the transformation of a metastable overlayer to the ground state structure for sufficiently thick overlayers.
Classification
Physics Abstracts
68.55 - 68.60 - 81.30K
1. Background.
The phenomenon of epitaxial growth has attracted continuous attention since the pioneering observa-
tions by Royer of the oriented growth of various
alkali halides on one another [1]. Since that time, considerable progress has been made from both the
experimental and theoretical point of view [2]. For example, it is understood that overlayer growth
occurs in one of three distinct modes : Frank-van der Merwe (FM) growth, Volmer-Weber (VW) growth,
or Stranski-Krastanov (SK) growth. In the first of
these, growth proceeds one monolayer at a time.
The growth is two-dimensional in the sense that no atoms occupy layer N until layer N - 1 is completely
filled. By contrast, VW growth occurs by the nuclea-
tion of solid clusters from the vapor phase that
condense as three-dimensional islands on the sub-
strate surface. The islands ultimately coalesce to
form a continuous film. The SK growth mode is a
combination of these two. A bulk overlayer forms by droplet nucleation after a few layers adsorb in layer- by-layer fashion. These growth modes are well
documented experimentally [3] and are also theoreti-
cally understood [4].
A much more challenging problem emerges if we next inquire into the detailed crystal structure of the growing epitaxial overlayer. We begin at the submo- nolayer level. Here, experiments generally reveal
ordered arrangements of overlayer atoms that
occupy well-defined adsorption sites defined by the
substrate [5]. One says that the adsorbate structure is commensurate with the substrate because the
overlayer lattice constant in each direction is a
simple rational multiple of the corresponding sub-
strate lattice constant (b). It is difficult to predict a
two-dimensional adsorbate crystal structure in this
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470120205500
regime because it depends in detail on the magnitude
of the (generally competing) weak interactions among adatoms at distances in excess of their
« natural >> lattice constant (a) [6], as well as on the
thermal motion of the adatoms.
A different type of competition becomes impor-
tant when the areal density of the adsorbate approa- ches one monolayer. Now short-range adatom-ada-
tom interactions favour an interparticle separation
of a, while adsorbate-substrate interactions favour
an interparticle separation of b. The so-called misfit, f = ( b - a ) /a quantifies the geometrical lattice
mismatch. A second parameter, /i, denotes the ratio of the adsorbate-substrate interaction strength to the
adsorbate-adsorbate interaction strength and descri- bes the chemical bonding mismatch at the interface.
A complete theory of monolayer epitaxy would provide a structural phase diagram as a function of
these two parameters and the temperature. At present, exact results are known only for a few important special cases. The most celebrated of these is the extension by Frank and van der Merwe [7] of a model originally introduced by Frenkel and Kontorova [8] to study slip and twinning in crystals.
A monolayer of foreign atoms interacting with a rigid substrate is modelled by a one-dimensional chain of atoms coupled by nearest-neighbour harmo-
nic springs subject to an external sinusoidal poten- tial. The period of the latter is taken as b, while the springs and between adatoms have natural extension
a. The spring constant and the potential well depth
determine the parameter A.
The Frank-van der Merwe model exhibits a num-
ber of characteristic features that are known to occur
in real epitaxial systems. If the misfit f is small (and
> is not too small), the overlayer strains uniformly to
match its lattice constant to that of the underlying
lattice. The overlayer is commensurate with the substrate. As the misfit increases, it eventually
becomes too costly to accomodate all the lattice mismatch through uniform strain. Instead, a particu-
lar type of defect - called a discommensura- tion - appears. A discommensuration is a narrow
region where the overlayer atoms are out of registry
with the minima of the substrate-induced periodic potential. Misfit is absorbed by a periodic array of discommensurations which separate regions of com-
mensurate registry. The defects first appear at a critical value of misfit, fc’ and their mean separation, 1, decreases as f increases so that there is
a smooth passage from the commensurate state to an
incommensurate state. The detailed nature of this structural phase transition is well understood from extensive studies of monolayer films of rare gas
atoms adsorbed onto lamellar substrates such as
graphite [9].
Now imagine continued deposition of foreign
atoms from the vapor phase so that growth proceeds
in a layer-by-layer FM mode. Suppose that f fc so
that the first monolayer is strained into commensura-
bility with the substrate. Succeeding overlayers will
lattice match to the chemically identical layer imme- diately below and hence also be in a state of uniform
strain. Van der Merwe [10] and later Jesser and Kuhlmann-Wilsdorf . [11] studied this problem with
an epitaxial model [12] that treats the substrate and
overlayer as adjacent elastic media with an external
sinusoidal stress imposed at the interface. One finds that there exists a critical film height he above which it costs too much energy to strain additional layers
into commensurability (or coherence) with the sub- strate. A fraction of the misfit again is accomodated
by a periodic array of defects so that a much reduced strain remains in the overlayer. In this case, the appropriate defect is called a misfit dislocation. In the simplest atomistic picture, one (edge) misfit
dislocation corresponds to either one extra row (if
a b) or one missing row (if a > b) of atoms in the
interfacial plane perpendicular to the direction of misfit. The misfit dislocation density increases as h increases until the average strain in the overlayer is
reduced to zero, i. e. , the lattice constant of the epitaxial layer is equal to its bulk crystal value. We
will always use the term coherent-incoherent transi- tion if we vary h and commensurate-incommensu-
rate transition if we vary f.
Qualitative tests of the theory of the coherent- incoherent transition sketched above have been made for a number of different substrate/overlayer
combinations [13]. For example, misfit dislocations may be directly observed with electron microscopy [14]. In metals where the lattice offers little resis- tance to dislocation motion, predicted values of he and the thickness dependence of the residual strain following loss of coherence are in rather good agreement with experiment [15]. By contrast, epi-
taxial combinations of oxides and semiconductors
generally exhibit he and residual strain values far in
excess of prediction [16]. It is believed that large
kinetic barriers to dislocation nucleation and migra-
tion in these systems prevent observation of the true
equilibrium structure. More recent progress in the
study of the coherent - incoherent transition has focused on introducing the true two-dimensional
structure of the epitaxial interface - particularly for
the case when the overlayer crystal structure differs
from that of the substrate, e.g., FCC (111) on
BCC (110) [17]. Coherence is lost through a compli-
cated interplay between arrays of misfit dislocations
along certain directions and uniform incoherent strains (« verniers ») along others. The final predic-
tions are in reasonable agreement with available data for metal-metal epitaxial combinations [18].
All of the work summarized above is relevant only
to the case of relatively small misfit ( f 10 %) .
The restriction arises from the explicit use of linear elasticity theory in the calculation of strain energies.
Indeed, the possibility that interfacial stresses at
large misfit could lead to mechanical instability of
the overlayer was pointed out quite early by Smollett
and Blackman [19]. This idea has been pursued [20]
with calculations of the ground state of a Frenkel-
Kontorova chain in which the harmonic interactions between neighbouring atoms are replaced by anhar-
monic forces. The authors interpret a breakdown of the model for large substrate-induced separations
between certain pairs of overlayer atoms as a signal
of the breakup of the chain into disconnected islands. While this might be relevant for some
systems, another possibility recently has attracted
considerable experimental attention : the overlayer adopts a crystal structure that is well lattice matched to the substrate but which differs from the crystal
structure that the overlayer material normally would adopt in the bulk. This phenomenon is known as pseudomorphism. The pseudomorphic phase is cohe-
rent with the substrate, or nearly so (small misfit),
whereas the bulk crystal structure would be highly
incoherent (large misfit). Pseudomorphism is
common in metal epitaxy as a consequence of the fact that many metals undergo structural transitions,
e.g., from FCC to BCC, both as a function of temperature and stress.
The growth of pseudomorphic epitaxial films was
first studied by vacuum evaporation of alkali halides in the early 1950s [21]. Renewed interest in this phenomenon accompanied the superior experimen-
tal control offered by molecular beam epitaxy growth techniques. For example, elemental tin crys- tallizes into its familiar body-centred tetragonal
« white tin » structure (a = 5.831 A, c = 3.181 A)
at room temperature. However, it adopts its low temperature « gray tin » diamond structure
(a = 6.489 A) when deposited at 300 K onto (001)
surfaces of InSb and CdTe (a = 6.48 A) [22]. The tin overlayer avoids the tremendous strain that would
occur in its normal metallic structure by epitaxial
stabilization of a crystal structure that appears elsewhere in its bulk phase diagram. Even more striking perhaps is the epitaxy of cobalt on gallium
arsenide [23]. Prinz showed that a GaAs(110) surface
can support epitaxial growth of BCC cobalt up to thicknesses approaching 103 A before the film trans- forms to its bulk HCP phase. The observed Co lattice constant (a = 2.819 A) matches very well to the semiconductor substrate ( f = 0.4 %) although a
BCC phase exists nowhere in the equilibrium phase diagram of cobalt. Subsequent electronic structure
calculations show that BCC cobalt is in fact a
metastable state which lies about 5 mRy above the ground state [24]. As a final example, we might cite
the epitaxial growth of superconducting Nb3Nb - (a = 5.25 A) on silicon [25] - an A15 compound
that does not exist in the bulk.
To our knowledge, there have been very few theoretical studies that address the question of epitaxial stabilization of metastable phases. On the
one hand, Machlin and Chaudhari [26] adopted a semi-empirical approach in an attempt to predict
candidate epitaxial pairs that might favor growth of
metastable overlayers. They suggested a general
« recipe » for such growth. On the other hand,
Jesser [27] investigated the first-order transformation of a metastable overlayer to its bulk stable phase using nucleation and elastic dislocation theory. He
obtained estimates of the size of critical droplets in good accord with observations of the FCC BCC transformation of iron epitaxially deposited on cop- per (100) [28].
2. Methodology and summary of results.
In this paper we present a theory of the structure of epitaxial bicrystals that is appropriate to the case
when the deposited material can adopt a metastable crystal structure that differs from its normal ground
state structure. Since the two competing phases can
differ substantially in both lattice constant and symmetry, our analysis includes the possibility of large misfit and structural transformation within the
overlayer. In particular, it is necessary to consider the response of an epitaxial layer to large shear
stresses in conjunction with strong adhesive forces from the substrate. Our basic approach involves a
nonlinear generalization of the Peierls-Nabarro model used by van der Merwe in his original investigations of the small misfit regime [10]. Accor- dingly, for most of our discussion, we restrict
ourselves to systems that adopt the two dimensional
layer-by-layer growth mode. Moreover, we will
allow only a single structural transition to occur as a
function of overlayer height. This is a significant
limitation since, for example, copper deposited epitaxially on W (110) is known to pass through a
sequence of structures during the initial stages of growth [29]. This case notwithstanding, we expect
our results to be most relevant to metal overlayers
since they are most likely to anneal to their equili-
brium structure on laboratory time scales.
A general description of the free energy density of
a bulk single crystal under large stress was given by
Cahn [30]. He noted that small strains (e) always
lower the symmetry of a crystal, while the associated deformation free energy density f B ( E ) increases proportional to E 2. This is simply the domain of
linear elasticity. However, as we increase the magni-
tude of the strain, one finds that (for special
directions of distortion) a new symmetry group becomes appropriate at one particular value e * of the strain. The symmetry at E * is higher than at
other points along the deformation path and the free energy density f B ( 8 * ) has a local minimum.
The structural transformation from BCC to FCC is an example of this phenomenon. A pure tetragonal
distortion of the BCC phase lowers the symmetry from cubic to tetragonal. But, since the BCC lattice
can equally well be regarded as face-centred tetrago- nal (Fig. 1), there is a special value of the strain, cla = J2, where we recover the cubic symmetry of
the FCC phase. Of course, this crystallographic correspondence does not address the energetics of
the transformation. To answer this question, Beau- champ and Villain [31] explicitly computed f B (e) using a number of empirical interatomic potentials appropriate for metals. In every instance, Cahn’s description was confirmed. The potential energy surface contains two local minima : one for FCC and
one for BCC. Furthermore, the lowest. energy trajec- tory through configuration space that connects the
two local minima in f B ( s ) almost exactly coincides
with the pure tetragonal deformation described above. In fact, we encounter the lowest energy barrier to transformation if the volume of the unit
Fig. 1. - Body-centred cubic cell (solid lines) with a face-
centred tetragonal cell delineated within (dashed lines).
cell fl is relaxed at every intermediate value of
c/a along the path of pure tetragonal strain. Hence,
there is a net volume change (albeit small) between
BCC and FCC. The path can be parametrized as
where 81 = log (cia) /log J2 and E2 E1 =
log (d2lno) - 3 o For BCC, E = E = 0 i and for
FCC, E1=1. The volume ratio between the BCC and FCC unit cells depends on the pair potential in question.
The free energy density as a function of El is
shown in figure 2. The two local minima at 81 = 0 and 81 = E1 are separated by a maximum at 8M that
is, in fact, a saddle point of the potential energy surface because any structure along the path of figure 1 is stable with respect to variations in the
remaining strain components. Let us parametrize
the curve in figure 2 with a fourth-order polynomial,
where A, B, and C are positive. If we set
T = 4 AC / B2, it is easy to verify that 81 = 0 is the lowest energy minimum if T > 1 while 81 = E* is the
lowest minimum if T 1. Consider now a straight-
forward application of the Landau theory of first order phase transitions [32] to the competition
between a stable FCC phase and a metastable BCC
phase [33] with 81 as the order parameter. Near the critical temperature, Tc, we are ied to precisely the
Fig. 2. - Free energy f B ( e 1 ) of intermediate crystal
structures as a function of tetragonal strain along the
lowest energy trajectory in configuration space. The path
connects a bulk stable cubic phase (E1 = 0) with a
metastable cubic phase ( El = E* . The points of mecha-
nical instability where a 2 f BI a Ei = 0 are denoted -E and
El. The intermediate structure with the greatest free energy occurs at e 1 = Em*
expansion set down in (2) for f B E1 . In this
phenomenological approach, the transformation
occurs through the temperature dependence of the
coefficients in (2) which we can express as
where « is a constant. Consequently, we regard figure 2 as a reasonably accurate representation of f B ( el) even for materials for which pair potentials
are not available because it is known that most solids favor the BCC phase at high temperature [34].
In the succeeding sections of this paper, we will examine the effect of epitaxial stress on the structural response of an overlayer material that possesses a deformation energy dependence in some « easy »
strain variable (not necessarily 61 as in the BCC- FCC case) that resembles figure 2. More precisely,
we generally will regard f B E1 times the film area
as the free energy density per unit height so that the
parameter h enters the energy as a multiplicative
constant as it does in the linear theories [10, 11]. Of
course, this is unlikely to be correct in detail when h is just two or three monolayers but, in the absence of
a more precise evaluation of f B in this limit, we will
retain this dependence and explore its consequences.
Our results are best understood in terms of the curvature of f B E1 . By definition, the curvature is positive in the vicinity of the local minima. Strained
crystals in these regions are stable. By contrast, the
vertical lines in figure 2 delineate a domain of strain where a2/BI ae; O. The bulk crystal is mechani-
cally unstable here ; it is impossible for a homoge-
neous external stress to force the system to assume a strain value within the interval IE-, El I. The
crystal fails or a discontinuous change of 61 occurs.
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