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Submitted on 1 Jan 1993
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Surface instabilities of epitaxial films on a substrate
N. Junqua, J. Grilhé
To cite this version:
N. Junqua, J. Grilhé. Surface instabilities of epitaxial films on a substrate. Journal de Physique III, EDP Sciences, 1993, 3 (8), pp.1589-1601. �10.1051/jp3:1993222�. �jpa-00249023�
Classification
Physics Abstracts
61.50C 68.55 68.60
Surface instabilities of epitaxial films on a substrate
N. Junqua and J. Grilh6
Laboratoire de Mdtallurgie Physique, U.A. 131 du C-N-R- S., Universitd de Poitiers, 40 avenue du Recteur Pineau, 86022Poitiers, France
(Receii'ed 20 January 1993, accepted J June J993)
Rdsum4, La mdthode des dislocations de surface est utilisde pour calculer la variation d'dnergie
d'un film mince en dpitaxie sur un substrat lorsqu'une rugositd de forrne sinusoidale apparait h sa surface. On ddterrnine une longueur d'onde critique ddpendant du rapport des coefficients
d'dlasticitd du substrat et du film et aussi de l'dpaisseur du film, au-dell de laquelle la variation
d'dnergie est ndgative. On discute dgalement de la cindtique du ddveloppement de la rugositd.
Abstract. The energy variation of an epitaxial film on a substrate is calculated when sinusoidal
roughness appears at the surface, using the method of the surface dislocation model. The energy variation is negative beyond a critical value of wavelength which depends on the thickness of the film and on the ratio of the shear moduli of substrate and film. The kinetic of roughness
development during film growing are discussed.
Introduction.
It has recently been observed [1-4] that a quasi-periodic thickness undulation of films or
multilayers epitaxially growing on a substrate can occur. The undulations appear for thicknesses lower than the critical thickness where the misfit dislocations appear and relax the strains. This mode of growth can be explained by a competitive effect between the surface energy and the elastic energy due to epitaxial misfit strains.
Several authors have undertaken different theoretical studies of the question. For instance, Srolovitz [5] uses a simple linear stability analysis, but does not account for the difference of elastic moduli between film and substrate. Spencer et al. [6] have performed a very complete study using continuum elastic theory to describe the stress state of the film/substrate system and have then applied transport mechanisms to explain the film growth.
These authors find that the flat surface of an epitaxial film is unstable with respect to
perturbations of wavelengths greater than a critical wavelength.
In the present study, the instability of the plane surface of a thin film on a substrate is
explained by considering a surface dislocation model which is applied to the cases of shear stresses and tensile or compressive stresses.
Principles involved.
When a film is grown epitaxially on a substrate in the absence of misfit dislocations, internal stresses, 30, appear in the crystals. The plane surface is free of stress, but when roughness is present relaxation stresses. 3~~j, appear which satisfy the equilibrium relationship :
(30 + 3~~j) n
=
0 (1)
where n is a vector normal to the surface oscillation.
The plane surface is the xoy plane the roughness is the sum of sinusoidal independent
fluctuations introduced in the Ox direction ; z~
= e~cos k~x (Fig. I), where the amplitude
e, is small compared to the wavelength A,. For each mode, with the amplitude e = e~, the
stability of the fluctuation can be studied by calculating the energy variation, AW (e, k, 30), of the system per unit area of the original surface. The energy variation can be expanded in
powers of e to the second order : Awie, k, 30)
= y AS (e, k) + AW~j~~(e, k, 30)
=
e~ A(30, k) + (2)
where y is the surface energy per unit area, AS and AW~j~~ are the surface variation and the elastic energy variation per unit area, respectively. The stability condition is then given by A(30, k)
~ 0.
The first term of AW is, to the second order in e
y AS (e, k)
= y l~ j~ VI + e~k~ sin~ kx dx ~
4 y
~~ ~~
(3)
A
o 4
The second term of AW can be calculated using the method of the surface dislocation model.
z vacuum
_~-
fflm
h e
substrate
Fig, I. Sinusoidal fluctuation of the free surface of a film on a substrate
z = e cos kx.
Surface dislocation model.
When the surface dislocation model is used, the free surface boundary conditions are satisfied by distributing dislocation arrays with suitable Burgers vectors at the surface of the crystal. The
stress field of these dislocations cancels out the relaxation stress field 3~~j at the surface and then it is possible to determine the distribution of each dislocation array.
Jagannadham et al. [7] have used this model for the calculation of stress and displacement
fields around edge and screw dislocations in a semi-infinite medium. They show that with this method the total stress components vanish outside the free surface of the finite body, unlike
with the image dislocation model. In many situations, merely placing an image dislocation does not cancel all of the surface stresses and therefore additional stress functions must be
imposed. Jagannadham et al. [8] use the surface dislocation model for the two-phase interface boundary problem where a lattice dislocation is situated in one of the phases : the interface is
planar and the semi-infinite media have different elastic moduli. The Airy stress functions for the two phases are then derived and are the same as those obtained by other methods,
This model facilitates the study of changes in the shape of the surface of a finite crystal when it is relieved of all surfaces stresses [9] and also permits the study of the stability of the
interface between two epitaxial crystals [10].
After the determination of the distributions of dislocation arrays that satisfy the free surface boundary conditions, the calculation of dislocations displacements, U~~j, is possible and elastic
energy AW~j~~ = U~~j Ion dS per unit area as shown by [9], calculated. Then the term
2 S
~
A(30, k) can be obtained.
The stress and energy calculations can be derived by solving the elasticity equations with the
boundary conditions of the problem, using the principle of the Green's function calculation by
Grinfield [I I] or by other methods [12], The surface dislocation model avoids the cumbersome mathematics involved in solving elasticity equations for the two-phase medium.
Roughness formation induced by shear stress,
The case of shear stress, «~ = To, induced by distortion between the lattice film of thickness ho and the lattice of the semi-infinite substrate is the easiest since only one screw surface
dislocations distribution I(x ) of Burgers vector (0, b, 0 is needed. For the calculation of each surface dislocation stress in the two phase medium, one uses a surface dislocation image as
shown for instance by Head [13]. The Burgers vector of a surface dislocation image due to the interface film/substrate is then (0, ah, 0) with a
=
(p'- p)/(p'+11) and the image
dislocations are at z =
2 ho from the surface film (p' and p are the shear modulus of the
substrate and the film).
To a first order in e, the equilibrium condition (I) of the free surface becomes
~~ ek sin kx + ~~
j~ ~~'~ dx' + tY ~~ j~ ~~~'~~~ ~'~
~
dX'
# 0
2 gr
~
x x' 2
gr _~
4 h( + (x x')
and I (x') is an odd function of x' I (x')
= c (ho) cos kx'. Calculation of the integrals [14] gives
the distribution
I(x)
= era
~ ~
~~ ~
cos kx.
~b[i + a e~ ° j
The free surface displacement is the sum of displacements induced by the distribution
I(x) and the image distribution. Since distribution is calculated for first order of e,
displacements are only calculated for zero order of e.
For each surface dislocation
~Y ~ ~ ~~ ~~
then for the distribution I(x), one obtains a displacement
U(~~~~ =
~~°
~ ~ ~
sin kx
ll i + a e~ °
For each image dislocation
u(~'~~ =
£ tan~ ~)£' + ~/
gr o
and then
eTo a 2hok
~~~ ~~
uimag
~
~ ~ , e
11 +
a e °
The total displacement is
U,, ix =
~~° " ~ ~~~~ sink-i.
/~ +
a e~ '°
The energy variation is obtained as shown by Grilhd [9] using
Then, the total energy variation to a second order in e is given by
~w
=
Ye~k~ i T/ i
a e [<[
j4~
4 p yk
~ ~ ~- ,i
and the critical wavelength corresponding to AW
=
0 is given by the relationship
A~ = ~~" =
~ ")~ ~ " ~ ~~~~,
~ T~ i a e u ~
For ho
- co, A~ - A~ =
2 grpy/r( the critical wavelength is the same as that for bulk material with the elastic modulus p of the film.
For ho - 0,
~ - A
~ =
2 gr p' y/T( which is the critical wavelength of bulk material with the elastic modulus p' of the substrate.
The limiting situations are obtained with r ~
=
~
=
~ "
equal to infinity or equal to 0
p a
for a rigid substrate, a
=
~
f k~ ~ 2h,j(
~
c k~ ~ 2hj~i~
~ ~~~h h
~k~
for a rigid film, a
=
~f kc I + e~ ~~°'~ l
A
c k~ ~- 2 Jiui~ tanh hok~
For the first limiting condition, a solution for instability is possible only if ho k~~ I.
The plane surface of the film on a infinitely rigid substrate is stable for any perturbation when the thickness ho is smaller than a critical thickness h~
ho ~h~
= =
~).
~~
12
'~~ ' r=
«
r-lo
to
r-s r-o.5
wsrABium r-o
8 I
i
~
i I
I
', I
4 ' ,
, ', I
', I
2 ', '
,
',
- ~~ SrAl1lUlY h~ /x~
o
~
o o-I o-1 03 0.4 o-S
Fig- 2. Critical wavelength JA~ as a function of the thickness ho/A of the film for various values of l~
= ll'Ill Case of shear stresses. A~/A~ varies from A~/A~ = r for ho =
0 to I for ho
- co.
These results are showed in the figure 2 ; the wavelength A~ corresponding to infinite ho is chosen as unity.
It is possible to obtain the most probable static fluctuation
k~ corresponding to
~()
=
0 with the equation
~ ~~ ~"l]~p ~~2/>jjlp j,
,
+ 4 «hok~
~ , =
2 ~
l + a e~ ~~° P (I + a e~ ?" P)~ ~t
The results are showed in the figure 3 and it can be noted than A~ m 2 A~ for all values of ho and L
Roughness formation induced by tensile or compressive stress.
This case has been studied by Spencer et al. [6] by other techniques.
The problem can be reduced to one intemal stress component, «),=«°, since
«), does not play a role in the relaxation when fluctuations in the Ox direction,
z = e cos kx, are studied.
The stress «), in an epitaxial thin film of thickness ho on a semi-infinite substrate is induced
by an elastic deformation in the layer definited by the relation data
= (a~ c~~)la~.
Two sets of surface edge dislocations are taken into account the first is the distribution
fix) with Burgers vector parallel to the surface and the second g(x) with Burgers vector
perpendicular to the surface.
2o
1
r -
j r-to
~~ l
~
l i
Ii
i
, ,
8 ',
, (
,
, (
'
, i
,
~ ' i
'
' ~
,
h~JX~
0
0 O-I O-1 03 0.4 o-S
Fig. 3. The most probable wavelength A/A as a function of the thickness ho/A of the film for various values of r
= p'/p ~ l. Case of shear stresses.
Equation ii now becomes
I+
w
«o ek sin kx + fix') [ek sin k,<«(, ix x') + « (, (x x')] dx' +
«
+ j~ ~ g(x')iek sin kx«t,(x x') + «t=(x x')i it' = 0 15)
and
+ «
f(x')[ek sin kx«),(x x') + «f,(x x')] dx' +
-«
+ «
+ g(x')[ek sin kx«f,(x x') + «§,(x x')] dx'
= 0. (6)
-«
The distributions can be developed as
f(-I " z e~ fn(X) et g(X
#
~ e" gn (X
Stress «( (x -.<') and «,( ix x') are obtained with Airy formulae for one dislocation. Airy
stress functions for an edge dislocation in a bimetallic medium have been obtained by Dundurs
jls, 16] and also by Jagannadham j8]. When the origin is taken on the dislocation at the free surface with the interface at z
=
h and with
~"i~+)K ~~~i>+I ~"~~~" ~")
~~ ~"qrj)~K)~4qr~~-v)
17)
(p and p', v and v' are the elastic constants of the film and the substrate), equations (5) and (6) can be written to a first order in e as
~
2ksinkx+ +w fi(x')x
K
_~
2 128 Ah((x x') 24 Ah((x x') (A + B )(x x> l~,
~ (x-x')~ (4h(+ (x-x')~)~ (4h(+ (x-x')~)~~ 4h(+ (x-x')~ ~
+ « 256 Ah( 48 Ah( 2(A B ho
+ gj lx')
~ ~ ~
+
~ ~ ~
+
~ ~
dx'
=
0 (8)
-w
(4 ho + (x x') ) (4 ho + lx x') 4 ho + (x x')
and :
+ « 256 Ah( 48 Ah( 2IA B ho
«
~ ~~'~
(4 h( + (x x' )~)~ (4 h( + (x x')~)~ 4 h( + (x x' )~
~' ~
+« 2 128Ah((x-x') 8Ah((x-x') (A B)(x-x>)I
+ gj(x')
,
+
~ ~ ~
+
~ ~ ~
+
~ ~ x
_~
X-X (4ho+ (X-X') (4ho+ (X-X') 4ho+ (X-X')
xdx'=0. (9)
It is clear from equations (8) and (9) that the array with Burgers vector parallel to the Ox axis is an odd function of.<', while the array with Burgers vector perpendicular to the Oz axis is an even function of x. Then
fj(x')
= c~(ho) cos kx' and g j(x') = b(ho) sin kx' Evaluation of the integrals [14] provides
+ ~
2 [A + B + 4 ho k(I + ho k)] e~ ~~°'
~~~°~
4 /~b ~° ~ [- (A + B + 4 Ah(k~ e~ ~°' + AB e~ ~ ~°' + l
~_~-4Ah(k~
~~~°~ ~ji "° ~ ~ ~~~
[- (A + B + 4Ah(k~ ~ ~~" ~ ~~ ~ ~~~ ~
The two distribution functions are now completely defined.
The displacement field of the f(x') and g (,<') distributions is now calculated in the film by
summations of each displacement u,(x -x', z =') (I
= x or z) of one dislocation over its
respective distribution.
+ w
U(
= ea (ho) u((x x',
z z') cos kx' da'
-«
I+
«
and Uf = eb (hu) ufix x', z z') sin kx~ dx'
«
The displacement field is calculated at a point (x, z) of the surface of the film. Since only
second order terms in e are needed in the energy calculation, the above expression for the energy shows that only zero order terms in e are needed in the expression for the displacements
u~ of each dislocation.
The field displacement of the distributions is then
uf
=
b ~~(h )~
' 2(1+ K)k ~
x [-4A(hok)~+2A(K + I)hok-AK -B- (K + I)e~~°'] e~~~°'sink.<
ug
=
b ~
b(h ) ~
' 2(1+ K)k °
x [4A(hok)~-2A(K I)hok-AK +B + (K I)e~~°']e~~~°'sinkx.
The energy variation per unit length is
AW ~'~~ 2 j~ (Uf + ug ~ ~~ ~i~ ~~ ~
~
' ' °
so, to a second order in e, the total variation energy (2) is then
l~
4 Ah 0 ~ ~ ~~~~~,~
~'~
4 Au
AW ~~
=
'~~~~ ~i~k "~
l (A + B + 4 ~ ~~° ~~~ ~ ~ ~ ~~ ~
As A increases, AW decreases and AW
=
0 for the critical wavelength A~ ; for ~A~,
AW is negative and perturbation is stable. Also when AW is at a minimum there is a most
probable wavelength A~.
The critical wavelength A~
=
2 gr/k~ is obtained using the expression
This
plain formulae is
available for all elastic
substrate (A and B are using (7)).
Figure 4
shows
A~ as a function of the of the film ho for
velength to infinite ho is chosen as unity ; this is the critical avelength
A~ of which
ppears at the surface of semi-infinite edium ith constants of
the
film
; when ho = 0, the ritical avelength is
of a emi-infinite medium with the
2gr
grpy 2gr grpy p'y
lo
/~ r-«
r-io r-s
' r-o-1
8 r-0
~
wsTABiLnY
6
i
4 i
', I
', I
2 ', I
','
~,
"] fi' SrABfIIW h~/~
0
0 0.1 0.2 03 0.4 0.5 0.6
Fig. 4. Critical wavelength AjA as a function of the thickness ho/A~ of the film for various values of r
= p'/p. Case of tensile or compressive stresses.
All the values of A~/A~ lie between A~/A~ = r
= p'/p for ho = 0 and A~/A~ = I for large ho. When the substrate is rigid, r
= co, A
=
I/K and B
= K ; when the film is rigid,
r
= 0, A
= I, B
=
I. These values can be substituted in the above equations to obtain the
curves for the limiting cases.
For the limiting case r
= co equation (11) becomes k~ K + 4 hok~e~~~°'-
K e~~~°~~
~S
K + (I + K~ + 4 hok~)e~~~°'+
K e~~~°~~
The second member of this equation as a function of hok~, varies as 4 hok~/(I + K ) for small hok~ to I for large hok~. There is a solution only if the slope of the first member
I/ho k~ is smaller than 4/(1 + K ). The instability is completely suppressed for film thicknesses less than a critical value given by
~~ ~~k~ ~jl'
The curves are similar to those obtained by Spencer [6] also the critical value thickness for
an infinitely rigid substrate is the same.
The figure 4 shows that a stiffer substrate has a stabilising effect on the plane surface of the film since the stability domain under a curve is larger when r
~ l than when r
~ l.
For r
~ l, the critical wavelength is always greater than ho. This is not the case when r ~ l, and it can be seen on the graph than only beyond a value of ho is the condition
A
~ ~ ho satisfied.
