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]


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


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,


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


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


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)


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( )


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=

>

the nonlinear effect of lattice (with figures) mismatch parameter on

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