the nonlinear effect of lattice (with figures) mismatch parameter on
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The Nonlinear Effect of Lattice Mismatch Parameter onMorphological and Compositional Instabilities of Epitaxial Layers
By
Chien H. Wu
ABSTRACT
The misfit deformation in a film-substrate system is mostly concentrated in the film when the
stiffness of the system is dominated by that of the substrate. Such is generally the case in
microelectronics applications. For many semiconductor materials with electronic properties
suitable for device applications, the associated mismatch parameter routinely falls in the range,
say, from -5% to +5%. Elastic strains of such magnitude provide a source of free energy for
configurational modifications. To examine such a possibility, the perturbation in elastic
deformation is first obtained in terms of a perturbation in a configurational variable/parameter.
This perturbation in elastic deformation and the misfit deformation are traditionally treated as
two separate infinitesimal deformations. As a result, the resulting stability condition is either a
function of the mismatch strain energy density, for the case of a morphological perturbation, or
completely unrelated to the underlying mismatch, for the case of a compositional perturbation.
In either situation, the sign of the mismatch is totally immaterial. In this paper, the perturbation
in elastic deformation is taken as a small deformation superimposed on the large mismatch
deformation in a nonlinear setting. The role of the mismatch is thus more fully explored and
uncovered.
Professor, Department of Civil and Materials Engineering (MC 246), University of Illinois atChicago, 842 West Taylor Street, Chicago, IL 60607-7023, (312)413-2644, Fax (312)[email protected]
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1. Introduction
When a thin film of Young's modulus E and Poisson's ratio is coherently grown on a thick
substrate of identical moduli, most of the deformation that is developed in the film-substrate
system is concentrated in the film. Let hs and as be, respectively, the thickness and lattice
parameter of the substrate. The film, if detached from the substrate, has a uniform stress-free
thickness h hf f= 0 and a constant lattice parameter a af f= 0 . Since the film is coherently grown
on the substrate, it actually carries a uniform biaxial strain that is equal to the lattice mismatch
parameter m s f f= ( ) /a a a0 0 , and a thickness strain that may be used to determine the actual
film thickness hf 0 in the film-substrate system. For many semiconductor materials with
electronic properties suitable for device applications, the associated mismatch parameter
routinely falls in the range, say, from -5% to +5%. Elastic strains of such magnitude provide a
source of free energy to affect configurational changes so as to alter the initially uniform
geometry defined by the constant film thickness hf 0 and constant lattice parametera f 0 . The film-
substrate system is said to be configurationally unstable at ( hf f0 0,a ) if a neighboring
configuration can be shown to posses a lower energy. The following configurational
perturbations are considered in this paper:
Morphological Perturbation. We begin by assuming that the stress-free film thickness hf
is actually of the form
h h h t Xf f= +0 1( ) cos , (1.1)
where 2 / defines a modulation wavelength in the X1 direction, and h t( ) is a thickness-
perturbation amplitude expressed as a function of time. The nonuniformity in hf , coupled with
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the uniform m , leads to an additional nonuniform elastic deformation, which, in turn,
contributes to the surface chemical potential that drives surface diffusion. It is this surface
diffusion equation that enables us to determine the sign of the exponent in the exponential
character of h t( ) . In this connection the nonuniform elastic deformation has been obtained as a
small perturbation from the uniform deformation m in a linearly elastic material (Asaro and
Tiller, 1972; Grinfeld, 1986; Srolovitz, 1989). We have recalculated in this paper this additional
elastic field as a small deformation superimposed on the finite deformation m in a general
nonlinear setting. Unlike the aforementioned linear elasticity result, which is independent of the
sign of m , our nonlinear result does depend on the sign of the lattice mismatch. The sign of the
mismatch has previously been shown to be of significance only when surface stress is present
(Wu, Hsu and Chen, 1998).
Compositional Perturbation. When the film is a binary alloy of uniform composition c0 ,
a configurational perturbation may be given by
c c A t X =0 1( ) cos , (1.2)
where A(t) is a composition-perturbation amplitude. With the perturbation, the original uniform
lattice parameter a f 0 becomes the nonuniform a f defined by
a af f A t X= +0 11( ( ) cos ) , (1.3)
where is the linear expansion coefficient per unit change in composition. The effect of the
above is an eigenstrain eeee (Cahn, 1961; Mura, 1982) defined by
eeee dddd = = = = , eSF( ) cos ,c c e X ASF0 113 3 , (1.4)
where eSF is a stress-free volumetric strain. This eigenstrain leads to an additional elastic field,
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which has been obtained as an elasticity solution independent of m (Cahn, 1961). Again, we
have recalculated this solution as a small deformation superimposed on the finite deformation
m in a nonlinear setting. While the previously known instability condition is totally
independent of m , the new result is a function of the lattice mismatch parameter.
A uniformly stressed uniform system may become deformationally unstable with respect to a
perturbation in deformation. The buckling of uniformly stressed elastic half-spaces (Biot, 1965)
and cracked infinite spaces (Wu, 1979, 1980) is an elastic instability that is closely related to the
above mentioned configurational instabilities. A typical result for such a nonlinear deformational
analysis reveals that a uniform tensile field has the ability to suppress a transverse perturbation in
elastic deformation, while a compressive field does just the opposite. The configurational-
instability results of this paper are the direct consequences of this nonlinear elastic phenomenon.
2. Finite Plane Elastostatics
In this section we outline the theory to be used, largely without derivation. Our exposition
follows closely that of our earlier paper (Wu, 1979). The deformation from XK to xk may be
represented by a transformation of the form
x x X X X u X XA A = = +( , ) ( , )1 2 1 2 , (2.1)
x X3 3 3= , (2.2)
where A is the Kronecker delta, and and A range from 1 to 2. The stretch ratios
( , , ) 1 2 3 define a state of uniform deformation, and u defines the superimposed small
deformation. We have arranged the coordinates in such a way that the X2 0= plane is the plane
of the film to be considered in the sequel. The deformation gradient F is
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F = =LNM
OQP
=
+
+
L
NMMM
O
QPPP
FF
u uu ukK
bB 00
00
0 03
1 1 1 1 2
2 1 2 2 2
3
, ,, , , (2.3)
where FbB is the 2x2 portion of FkK . We shall need the following invariants:
J J FbB= = =det , det( )F 3 0 J 0 , (2.4)
I F F I F FiI iI bB bB= = + =0 32 , I0 . (2.5)
We take the material to be isotropic with elastic energy density given by
W( ) ( ) ( ) ( ) ( )F = + + + + + 2
3 1 1 11 2 32
12
22
32 , (2.6)
where ( , , ) 1 2 3 are the principal stretch ratios, and and are the Lame constants of
linear elasticity. We adopt this nonlinear form to accommodate the large misfit strain and also
for the fact that it actually tends to the elasticity energy density as I 1 (I = 1,2,3) . Setting
3 3= in (2.6), we obtain, for the case of generalized plane strain,
W H R J= 2 0 3 0 ( , ) , (2.7)
R I J r u u0 0 01 2
1 1 2 2 22= + + + + +( ) ( , , ) ,/ L r = 1 , (2.8)
J u u u u u u0 1 2 1 2 2 2 1 1 1 1 2 2 1 2 2 1= + + + ( , , ) , , , , (2.9)
H R R R( , )( )
( ) ( )0 3 02 3
0 32
322
41
32
1 12
14
3
= + + LNMOQP + + + LNM
OQP . (2.10)
In terms of the strain energy density W, and hence the function H, the components of the Piola
stress tensor are
P WF
H RR
FH R
RFA
AA AB B
=
=
+
LNM
OQP
RSTUVW
2 10 30
0 3
0
( , ) ( , ), (2.11)
where and AB are the two-dimensional alternator, and H H R/ 0 . The two
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displacement equations of equilibrium are
u rHH r
u
,
( , ), ,+
LNM
OQP = 3 1 0 H =
+ 22
, (2.12)
which is in complete agreement in form with the governing equations of linear elasticity. It
follows that the convenient complex-variable formulation applies.
To this end, we introduce complex variable Z X iX= +1 2 and complex displacement
u Z) u X X iu X X( ( , ) ( , )= +1 1 2 2 1 2 . (2.13)
Then, in terms of holomorphic functions (Z) and (Z) , we have
u Z) Z) Z Z) Z)( ( ( (= , (2.14)
where
( , ) ( , ) / ( , )r rH H r rH H r3 3 3= + . (2.15)
The Piola stress components may be combined to form the complexvariable expressions:
P iP P P iP iP P P iP22 21 22 22 21 11 12 11 11 12 = + + = + +$ ( & & ), $ ( & & ) P , (2.16)
where
$ / ( , ) , $ / ( , )P H r P H r22 3 1 11 3 22 2 = = , (2.17)
( & & ) /P iP Z22 21 2 = + + + , (2.18)
( & & ) /P iP Z11 12 2+ = + , (2.19)
in which
( , ) ( , ) ( , ) / ( , )r H r H rH H r rH H r3 3 3 34= . (2.20)
Finally, along a curve C defined by
Z C L) iC L)= +1 2( ( , (2.21)
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where L measures the arc length, the Piola traction P iP1 2+ is
P
iP P dC
dLi
P dCdL
i ddL
Z) Z Z) Z)1 2 11 2 22 12 2 2 2
+ = + +$ $
( ( ( . (2.22)
3. Morphological Instability
The thickness hf of an epitaxially deposited film is assumed to have the modulation defined by
(1.1), where h t hf( )
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where X L C L)1 1= = ( and A = C2 , in the notation and sign convention of (2.22). The
traction-free condition along the surface (3.3) may now be obtained by substituting the above into
(2.23). Thus,
$ ( , )P H22 1 2 3 10 0= + = , (3.4)
( ( ( $ cosZ) Z Z) Z) i P h X+ + = 11 12 , (3.5)
where Z is given by (3.3), and
$ / ( , )P H11 1 2 3 2 1 22 = + = (3.6)
by virtue of (3.4). By applying an analytic continuation to (3.5), we may satisfy (3.5) by
expressing both and in terms of a single function 0 as follows:
( ( ( ( )Z) Z), Z) Z= 0 0 (Z) = -Z 0 . (3.7)
0 1111
4 04 0
(( $ / )
( $ / ).Z)
P h ei P h e
i Z
i Z=
>