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Surface strains and measurements of misfit dislocation density by diffraction methods in thin films on substrates
Thin Solid Films, 250 (1994) 37-41
37
Surface strains and measurements of misfit dislocation density by diffraction methods in thin films on substrates N. Junqua and J. Grilh6 Laboratoire de Mdtallurgie Physique, URA 131 du CNRS, Universitb de Poitiers, 40 avenue du Reeteur Pineau, 03) 86022 Poitiers, France
(Received December 3, 1993; accepted April 7, 1994)
Abstract In-plane strain variations near the free surface of an epitaxial thin film containing misfit dislocations have been calculated. It is shown that strain distribution under the surface of the film is not constant and this distribution also depends on film thickness even for constant misfit dislocation density. The strain derived from diffraction measurements must therefore be corrected to obtain the true misfit dislocation density.
1. Introduction Stresses in films epitaxially grown on a substrate are often measured by surface diffraction methods with electrons, X-rays or neutral atoms. Fast reflection high energy electron diffraction ( R H E E D ) is generally performed in s i t u in molecular beam epitaxy (MBE) experiments and allows the evolution of strains and stresses during crystal growth to be followed [1, 2]. At the beginning of the deposition the measured surface parameter a is equal to the substrate parameter as and strain and stress in the film can be easily deduced. For isotropic films, it is known [3] that e = exx = ey>, = ~ao/a~ and cr = a x x = a~y = E B e , where 6a o = a r - a s is the lattice misfit and a~ and E B are the lattice parameter and elastic biaxial modulus of the film. When the film thickness h increases, the stresses are progressively relaxed and the surface parameter a goes to at. This relaxation begins when h becomes larger than a critical value hc, first calculated by Matthews and coworkers [4, 5], and corresponds with the appearance of misfit dislocations of Burgers vector Ibl = as at the film-substrate interface. The dislocation density p, or the mean distance Xo between misfit dislocations (p = b / x o ) , is frequently deduced by comparing the measured surface parameter a with its arithmetic average aav :
to deduce p from a and h. In the first part, the strain exx(r) and the displacement u(r) are calculated at the free surface. In the second part, the positions of diffraction peaks are deduced when only the free surface is diffracting. Then, in a third part, the contribution of planes under the free surface is presented.
2. Strains and displacements at the free surface The origin of the coordinate system is placed at a misfit dislocation at the film-substrate interface. The Oz axis is perpendicular to the surface. Only one set of misfit dislocations, parallel to the Oy axis and with a Burgers vector b (as, 0, 0) is considered (Fig. 1). The elastic constants of the film and the substrate are assumed to be of the same order and their differences are neglected. The results can be easily extended when this hypothesis is not fulfilled.
M(x,z) film ...L_
_2_
_J_
as a ~aav-
1 -(+as/Xo)
(1)
where + is the sign of 6ao. However, surface diffraction does not give an arithmetic average. In this paper we will show that, for a given dislocation density, the measured surface parameter depends on the film thickness h. A method is given
0040-6090/94/$7.00 SSDI 0040-6090(94)06162-E
Fig. 1. Array of misfit dislocations in an epitaxial thin film.
© 1994 - - Elsevier Science S.A. All rights reserved
N. Junqua, J. GrilhO / Measurements of misfit dislocation density
38
Stress components of a dislocation near a free surface have been calculated by Head [6]. For an edge dislocation, an image dislocation and an additional term must be considered. I f the edge dislocation is located at x = x ' and z = 0, the stress components at the point (x, z = h) of the free surfaces are 0". 2
~
• , ,
-1.130
(X
--
X')
',
.
, ~
/
/
-
,, ,
,'
',,'.
; ,,
-2.50 -
h/x o = I
-3.00
--
h/x o = 0.5
.....
h/Xo = 0.1
. . . .
h/x o = 0.05
I
I
[
0.2
0.4
0.6
0.8
1 XJX o
2rc_h cosh( 2r& /Xo)COS( 2nX /Xo) - 1 Xo ( Xo [cosh(2rch/xo) - cos(2zrX/Xo)] 2
4=D
sinh(2rch/xo)
-
--
0
Using summation methods described by Hirth and Lothe [7, 8] we obtain
~;
- [ c o s h ( 2 r r h ~ o) ------c-~s(os(2nX/Xo)[ J
(3)
The strain component exx is easily deduced from the relationship 1 __y2
Zxx(x, h)
(4)
SO
exx(X, h) = b ~2rch cosh(2r&/Xo)COS(2~X/Xo) - 1 Xo ( Xo [cosh(2nh/xo) - c o s ( 2 ~ z X / X o ] 2
-
~
, ,'
-3.50
E
/
-2.00
+~ (x - nXo) ~ Zxx(X, h) = - 8 h D ,,=~'-~o [h2 + (x - nxo) 2] 2
exx(X, h) -
. . •
2
x ' ) 2] 2
with D = Eb/4g( I - v2). E is the Young modulus and v the Poisson ratio. The stresses Zu(x, h) induced by a set of misfit dislocations at the point x of the free surface are obtained by summing eqns. (2), with x ' = nx o. For example, the Zxx(X, h) component is equal to
Sxx(X, h) =
° - . . . . .
,
-I.50
(2)
- 8 h [h 2 + ( x -
"
i
;
~xx -
-0,50
0
o'x.. ----0 - -D
6 a/a - 0.00 b/x o
sinh(27~h/xo ~; Lcosh(2rth/-~o) Z ~os(2~X/Xo)lJ
Other stress and strain components can be obtained in the same way. As they are not needed in the following, they have not been calculated. The relative variation of the surface lattice parameter 6a(x)/a~ = [a(x) -a~l/a~ is equal to ex.,.(x, h). It is plotted, divided by b/xo, in Fig. 2 for various values of h/xo. Starting from h = 0 we can explain the evolution of these curves. For this case, a(x) = a~ and 6a is zero everywhere. At the beginning of the film growth, i.e. for h = 0.05Xo, a strong relaxation occurs very localized near dislocations. As is well known for edge dislocations, the strain is m a x i m u m in planes at an angle of + ~/2 from the glide plane. This is slightly modified but
Fig. 2. Surface parameter variations for different thicknesses h.
remains roughly true for dislocations near a surface• This explains why the strain relaxation at the free surface is concentrated in narrow regions located at x ~ n x o + h (n an integer). As h increases (h = 0.1Xo), the relaxation strain at the free surface becomes less intense and relaxation regions spread and collapse (h ~ 0.5Xo) (Fig. 2). For h ~> Xo, the surface deformation ex.,.(x, h) becomes constant. The displacement vector u(x, h) at the free surface can be deduced from the strain tensor with the classical formula [9]. F o r example, the component ux(x, h), needed in the next part to calculate diffraction intensities, is given by the integral Ux(X, h) - Ux(0, h) - 1 --E v 2 £x Xx.~(x', h) dx' or after integration and with ux(0, h) = 0
u~(x, h) "
sin( 2ztX /Xo)
b ~h
= ~ [7~-ocosh(2rch-/~o) ~ c--~-s(2rtxlxo)
I -tan-J
sinh(g~zh/x°) (rc2)]} cosh(2~zh/xo) - 1 tan
(5)
This relationship gives only the displacements induced by the misfit dislocations at the free surface of the epitaxial film. To obtain the total elastic displacement, the deformation u°(x) = x fia/a due to the epitaxy without misfit dislocations must be added: Ux(x, h ) = u x ( x , h ) + u ? ( x )
N. Junqua, J. Grilhb/ Measurementsof misfit dislocationdensity 3. Diffracted intensity calculations and measures of surface strains
39
1
i
If a beam of particles (electrons, neutrons or photons) is sent onto the free surface atoms, the diffracted wave amplitude ~,(k) is given by [10, 11] 4'(k) = C ~ exp[2rd(k - ko) • (rj + Uj)] J where ko and k are the incident and diffracted wavenumbers, Uj is the displacement vector of the j t h atom of the surface located at r = rj and C is a constant depending on the apparatus. As a first approximation, only the atoms of the free surface are considered. The extension of the model for diffraction by a thicker layer under the surface is discussed later. The diffracted intensity, equal to O*0, is then proportional to
l(k) = 0~b * ~
exp[2rci(k - ko) • (rj +
10
0
40
x0k&
50
h/xo
Ak • (rj + Uj) ~ Akx[x j + Ux(xj, h)]
adiff/a s 0.98
•
•
ep = 1 / 200o
•
rap= Ii lOOm
•
* p=l/40*
*
0.98
#
0.96
0.96
The diffracted intensity I(Akx) is then equal to
AA&•Ajdt&,LIt •
•
p= i120 4
•
)2
l ( a k d ~ ( ~ = cos{2~ ~dcx[x; + U~(xj, h)]}
(x~a
0.94
0.94
0.92
0.92
)2 sin{2z Akx[xj + U,(xj, h)]}
+
30
Fig. 3. Simulated diffraction intensity for a thin film of thickness = 10 .3 and a misfit dislocation density p = 1/20.
In such experiments ( R H E E D measurements, for example) the angle between the incident wave and the surface is very small ( ~ 1-3 °) [l, 12]; the diffraction of electrons occurs within a very thin surface layer of the crystal which can be idealized as a two-dimensional object; thus the following approximation can be made for diffracting planes perpendicular to the surface and with k - ko = Ak:
~a/x°/a
20
(6)
~/=1
where ux(x;, h) is given by eqn. (5) thus l(Ak~) can easily be determined from eqn. (6). Figure 3 shows an intensity curve I(Ak~) in k space as a function of the dimensionless parameter Xo Ak~ for a thickness h/xo = 10 -3 . This curve can be directly compared with an intensity curve obtained in a diffraction experiment. The value (Xo Akx)max between two intensity maxima defines an "average diffracted" lattice parameter ad~fr/ Xo = 1/(Xo Ak~) . . . . or adiff/as = 1/p(x o Akx)ma x (p is the density of misfit dislocations) different from the arithmetic average a~v given by eqn. (1). adler, which is the lattice parameter deduced by diffraction experiments, depends on the film thickness for a given misfit dislocation density. The number p of deposited planes can be expressed as p = h la~<,.= h lpxo. The variations of ad~n-la~ are reported in Fig. 4 for different dislocation densities. The calculations were performed with af ~ as, i.e. for tensile stresses in the film. For af/> as, i.e. for compres-
++Jr++ + + 0.9
++++
' 10
+ p=l/10+ '
20
+ '
'
0.9
30 40 50 number of deposited planes
Fig. 4. Evolution of the surface parameter adi ff deduced f r o m t h e m a x i m a of the diffracted intensity curves as a function of the n u m b e r of deposited planes and for different dislocation densities.
sive stresses in the film, the strain is obtained by replac-ing b by - b in eqns. (3) and (4) and also in eqn. (5) to obtain the displacement; then adifr(h) can be deduced from Fig. 4 using the symmetry about adiff/ a s = 1. It should be noted that for a large number of deposited planes adifr/a s-..+ aav/as or af/as when the epitaxial dislocations relax all the lattice mismatch between the substrate and the film.
NI Junqua, J. Grilhb / Measurements of misfit dislocation density
40
4. Contribution of the planes under the surface to the diffracted intensity In the preceding section we assumed that only the atoms at the free surface are diffracting. The validity of this hypothesis depends on the absorption length of the incident wave. In R H E E D experiments, the grazing incidence and detection angles mean that a long mean free path through the sample is associated with penetration normal to the surface of only a few atomic layers. We used the approximation that, for the fifth diffraction plane, the intensity becomes divided by 100, then the contribution of the j t h plane under the surface to the diffracted intensity is obtained with the attenuation factor exp( - j a / ~), ~ ~ a, and is negligible for j > 4. Calculations of the strain in planes under the free surface were performed. The stress component axx(x, z, h) of the dislocation in a semi-infinite solid is obtained by superposition of the stress fields of the dislocation and the image and with the supplementary term as shown by Head [5]. Summing the contributions of the dislocations on a row and using eqn. (4), we obtain the expression for the strain ex.,.(x, z, h) for dislocations at a distance xo. The analytic form of exx(x, z, h) is very large so only an example of exx(X) in the four planes near the free surface is shown in Fig. 5 when the density of epitaxial dislocations is p = 1/10 and the number of deposited planes is five. The displacement ux(x, z, h) on planes under the surface is calculated and summing eqn. (6) with the above attenuation coefficient we obtain the corresponding intensity curve I(Ak~). -0.7
i
a/a b/xo
a
i
9=1/20 p=l/lO
0.98
12 four diffracting planes 0.96
0.96 •
adi ff/a s
A
•
/~
0.94
0.94
0.92
?
0.92
: ~
•
0.9
• l
10
20
[]
•
[]
i
I
30 40 number of deposited planes
0.9 50
Fig. 6. E v o l u t i o n o f the p a r a m e t e r adi ff as a f u n c t i o n o f the n u m b e r o f d e p o s i t e d p l a n e s f o r d i s l o c a t i o n densities p = 1/10 a n d p = 1/20 in the f o l l o w i n g t w o cases: m , A , s u r f a c e p l a n e o n l y diffracting; ~ , L , f o u r d i f f r a c t i n g planes.
The variations of the parameter adiff for p = 1/10 and p = 1/20 are presented in Fig. 6 when only the surface plane is diffracting and when the contribution to the diffracted wave of four atomic planes under the surface is accounted for. The figure confirms that the contribution of the planes under the surface is weak for the R H E E D experimental conditions.
5. Conclusion
i
8
In epitaxial thin films with misfit dislocations, the surface strain presents strong variations for small film thicknesses (h << Xo). This inhomogeneity decreases as h increases and disappears for h ~ xo. One of the main consequences is that the surface parameter adi~ measured by diffraction methods is different from aav and varies with h for a constant dislocation density. The direct comparison of measured and computed intensity diffraction, or the comparison between the surface parameters ad~frdeduced from these curves (Fig. 4), gives a method of obtaining the real dislocation densities from such experiments. These corrections are needed only when h < Xo.
-- - surface .... plane 1 - - - plane 2 plane 3
-0.8 "
•
•
0.98
-0.9
-1.t
Acknowledgments -1.2 0
I 0.2
t 0.4
I 0.6
I 0.8 x/x
Fig. 5. P a r a m e t e r v a r i a t i o n s in p l a n e s u n d e r t h e free s u r f a c e f o r p = 1/10 a n d a film t h i c k n e s s h = 5a.
The authors would like to thank M. Piecuch and V. Pierron-Bohnes for helpful discussions and comments and M. Small for reading the manuscript and making useful suggestions.
N. Junqua, J. Grilhk / Measurements of misfit dislocation density
References 1 Ch. Chatillon and J. Massies, in FundamentalPhenomena in Nucleation and Growth of Crystals, Materials Science Forum, Vols. 59-60, Trans Tech, Aedermannsdorf, Switzerland, 1990, pp. 230-285. 2 Ph. Oudy, in Fundamental Phenomena in Nucleation and Growth of Crystals, Materials Science Forum, Vols. 59-60, Trans Tech, Aedermannsdorf, Switzerland, 1990, pp. 581-616. 3 W. D. Nix, Mechanical properties of thin films, Metall. Trans. A, 20 (1989) 2217-2245. 4 J. W. Matthews, J. Vac. Sci. Technol., 12 (1975) 126. 5 J. W. Matthews, A. E. Blakeslee and S. Mader, Thin Solid Films, 33 (1976) 253. 6 A. K. Head, Philos. Mag., 44 (1953) 92.
41
7 J. P. Hirth and J. Lothe, Theory and Dislocations, Wiley, New York, 1982. 8 P. M. Morse and H. Feshback, Method~ of Theoretical Physics, Vol. 1, McGraw-Hill, New York, 1953, p. 413. 9 S. Timoshenko and J. N. Goodier, Theory of Elasticity, McGrawHill, New York, 1951, p. 32. 10 A. Guinier, Thborie et Technique de la Radiocristallographie, 3rd edn., Dunod, Paris, 1964. 11 M. Piecuch and L. Nevot, in Fundamental Phenomena in Nucleation and Growth of Crystals, Materials Science Forum, Vols. 59-60, Trans Tech, Aedermannsdorf, Switzerland, 1990, pp. 93139. 12 H. Hfith, Surfaces and Interfaces of Solids, 2nd edn., Springer, New York, 1993, pp. 201-209.
37
Surface strains and measurements of misfit dislocation density by diffraction methods in thin films on substrates N. Junqua and J. Grilh6 Laboratoire de Mdtallurgie Physique, URA 131 du CNRS, Universitb de Poitiers, 40 avenue du Reeteur Pineau, 03) 86022 Poitiers, France
(Received December 3, 1993; accepted April 7, 1994)
Abstract In-plane strain variations near the free surface of an epitaxial thin film containing misfit dislocations have been calculated. It is shown that strain distribution under the surface of the film is not constant and this distribution also depends on film thickness even for constant misfit dislocation density. The strain derived from diffraction measurements must therefore be corrected to obtain the true misfit dislocation density.
1. Introduction Stresses in films epitaxially grown on a substrate are often measured by surface diffraction methods with electrons, X-rays or neutral atoms. Fast reflection high energy electron diffraction ( R H E E D ) is generally performed in s i t u in molecular beam epitaxy (MBE) experiments and allows the evolution of strains and stresses during crystal growth to be followed [1, 2]. At the beginning of the deposition the measured surface parameter a is equal to the substrate parameter as and strain and stress in the film can be easily deduced. For isotropic films, it is known [3] that e = exx = ey>, = ~ao/a~ and cr = a x x = a~y = E B e , where 6a o = a r - a s is the lattice misfit and a~ and E B are the lattice parameter and elastic biaxial modulus of the film. When the film thickness h increases, the stresses are progressively relaxed and the surface parameter a goes to at. This relaxation begins when h becomes larger than a critical value hc, first calculated by Matthews and coworkers [4, 5], and corresponds with the appearance of misfit dislocations of Burgers vector Ibl = as at the film-substrate interface. The dislocation density p, or the mean distance Xo between misfit dislocations (p = b / x o ) , is frequently deduced by comparing the measured surface parameter a with its arithmetic average aav :
to deduce p from a and h. In the first part, the strain exx(r) and the displacement u(r) are calculated at the free surface. In the second part, the positions of diffraction peaks are deduced when only the free surface is diffracting. Then, in a third part, the contribution of planes under the free surface is presented.
2. Strains and displacements at the free surface The origin of the coordinate system is placed at a misfit dislocation at the film-substrate interface. The Oz axis is perpendicular to the surface. Only one set of misfit dislocations, parallel to the Oy axis and with a Burgers vector b (as, 0, 0) is considered (Fig. 1). The elastic constants of the film and the substrate are assumed to be of the same order and their differences are neglected. The results can be easily extended when this hypothesis is not fulfilled.
M(x,z) film ...L_
_2_
_J_
as a ~aav-
1 -(+as/Xo)
(1)
where + is the sign of 6ao. However, surface diffraction does not give an arithmetic average. In this paper we will show that, for a given dislocation density, the measured surface parameter depends on the film thickness h. A method is given
0040-6090/94/$7.00 SSDI 0040-6090(94)06162-E
Fig. 1. Array of misfit dislocations in an epitaxial thin film.
© 1994 - - Elsevier Science S.A. All rights reserved
N. Junqua, J. GrilhO / Measurements of misfit dislocation density
38
Stress components of a dislocation near a free surface have been calculated by Head [6]. For an edge dislocation, an image dislocation and an additional term must be considered. I f the edge dislocation is located at x = x ' and z = 0, the stress components at the point (x, z = h) of the free surfaces are 0". 2
~
• , ,
-1.130
(X
--
X')
',
.
, ~
/
/
-
,, ,
,'
',,'.
; ,,
-2.50 -
h/x o = I
-3.00
--
h/x o = 0.5
.....
h/Xo = 0.1
. . . .
h/x o = 0.05
I
I
[
0.2
0.4
0.6
0.8
1 XJX o
2rc_h cosh( 2r& /Xo)COS( 2nX /Xo) - 1 Xo ( Xo [cosh(2rch/xo) - cos(2zrX/Xo)] 2
4=D
sinh(2rch/xo)
-
--
0
Using summation methods described by Hirth and Lothe [7, 8] we obtain
~;
- [ c o s h ( 2 r r h ~ o) ------c-~s(os(2nX/Xo)[ J
(3)
The strain component exx is easily deduced from the relationship 1 __y2
Zxx(x, h)
(4)
SO
exx(X, h) = b ~2rch cosh(2r&/Xo)COS(2~X/Xo) - 1 Xo ( Xo [cosh(2nh/xo) - c o s ( 2 ~ z X / X o ] 2
-
~
, ,'
-3.50
E
/
-2.00
+~ (x - nXo) ~ Zxx(X, h) = - 8 h D ,,=~'-~o [h2 + (x - nxo) 2] 2
exx(X, h) -
. . •
2
x ' ) 2] 2
with D = Eb/4g( I - v2). E is the Young modulus and v the Poisson ratio. The stresses Zu(x, h) induced by a set of misfit dislocations at the point x of the free surface are obtained by summing eqns. (2), with x ' = nx o. For example, the Zxx(X, h) component is equal to
Sxx(X, h) =
° - . . . . .
,
-I.50
(2)
- 8 h [h 2 + ( x -
"
i
;
~xx -
-0,50
0
o'x.. ----0 - -D
6 a/a - 0.00 b/x o
sinh(27~h/xo ~; Lcosh(2rth/-~o) Z ~os(2~X/Xo)lJ
Other stress and strain components can be obtained in the same way. As they are not needed in the following, they have not been calculated. The relative variation of the surface lattice parameter 6a(x)/a~ = [a(x) -a~l/a~ is equal to ex.,.(x, h). It is plotted, divided by b/xo, in Fig. 2 for various values of h/xo. Starting from h = 0 we can explain the evolution of these curves. For this case, a(x) = a~ and 6a is zero everywhere. At the beginning of the film growth, i.e. for h = 0.05Xo, a strong relaxation occurs very localized near dislocations. As is well known for edge dislocations, the strain is m a x i m u m in planes at an angle of + ~/2 from the glide plane. This is slightly modified but
Fig. 2. Surface parameter variations for different thicknesses h.
remains roughly true for dislocations near a surface• This explains why the strain relaxation at the free surface is concentrated in narrow regions located at x ~ n x o + h (n an integer). As h increases (h = 0.1Xo), the relaxation strain at the free surface becomes less intense and relaxation regions spread and collapse (h ~ 0.5Xo) (Fig. 2). For h ~> Xo, the surface deformation ex.,.(x, h) becomes constant. The displacement vector u(x, h) at the free surface can be deduced from the strain tensor with the classical formula [9]. F o r example, the component ux(x, h), needed in the next part to calculate diffraction intensities, is given by the integral Ux(X, h) - Ux(0, h) - 1 --E v 2 £x Xx.~(x', h) dx' or after integration and with ux(0, h) = 0
u~(x, h) "
sin( 2ztX /Xo)
b ~h
= ~ [7~-ocosh(2rch-/~o) ~ c--~-s(2rtxlxo)
I -tan-J
sinh(g~zh/x°) (rc2)]} cosh(2~zh/xo) - 1 tan
(5)
This relationship gives only the displacements induced by the misfit dislocations at the free surface of the epitaxial film. To obtain the total elastic displacement, the deformation u°(x) = x fia/a due to the epitaxy without misfit dislocations must be added: Ux(x, h ) = u x ( x , h ) + u ? ( x )
N. Junqua, J. Grilhb/ Measurementsof misfit dislocationdensity 3. Diffracted intensity calculations and measures of surface strains
39
1
i
If a beam of particles (electrons, neutrons or photons) is sent onto the free surface atoms, the diffracted wave amplitude ~,(k) is given by [10, 11] 4'(k) = C ~ exp[2rd(k - ko) • (rj + Uj)] J where ko and k are the incident and diffracted wavenumbers, Uj is the displacement vector of the j t h atom of the surface located at r = rj and C is a constant depending on the apparatus. As a first approximation, only the atoms of the free surface are considered. The extension of the model for diffraction by a thicker layer under the surface is discussed later. The diffracted intensity, equal to O*0, is then proportional to
l(k) = 0~b * ~
exp[2rci(k - ko) • (rj +
10
0
40
x0k&
50
h/xo
Ak • (rj + Uj) ~ Akx[x j + Ux(xj, h)]
adiff/a s 0.98
•
•
ep = 1 / 200o
•
rap= Ii lOOm
•
* p=l/40*
*
0.98
#
0.96
0.96
The diffracted intensity I(Akx) is then equal to
AA&•Ajdt&,LIt •
•
p= i120 4
•
)2
l ( a k d ~ ( ~ = cos{2~ ~dcx[x; + U~(xj, h)]}
(x~a
0.94
0.94
0.92
0.92
)2 sin{2z Akx[xj + U,(xj, h)]}
+
30
Fig. 3. Simulated diffraction intensity for a thin film of thickness = 10 .3 and a misfit dislocation density p = 1/20.
In such experiments ( R H E E D measurements, for example) the angle between the incident wave and the surface is very small ( ~ 1-3 °) [l, 12]; the diffraction of electrons occurs within a very thin surface layer of the crystal which can be idealized as a two-dimensional object; thus the following approximation can be made for diffracting planes perpendicular to the surface and with k - ko = Ak:
~a/x°/a
20
(6)
~/=1
where ux(x;, h) is given by eqn. (5) thus l(Ak~) can easily be determined from eqn. (6). Figure 3 shows an intensity curve I(Ak~) in k space as a function of the dimensionless parameter Xo Ak~ for a thickness h/xo = 10 -3 . This curve can be directly compared with an intensity curve obtained in a diffraction experiment. The value (Xo Akx)max between two intensity maxima defines an "average diffracted" lattice parameter ad~fr/ Xo = 1/(Xo Ak~) . . . . or adiff/as = 1/p(x o Akx)ma x (p is the density of misfit dislocations) different from the arithmetic average a~v given by eqn. (1). adler, which is the lattice parameter deduced by diffraction experiments, depends on the film thickness for a given misfit dislocation density. The number p of deposited planes can be expressed as p = h la~<,.= h lpxo. The variations of ad~n-la~ are reported in Fig. 4 for different dislocation densities. The calculations were performed with af ~ as, i.e. for tensile stresses in the film. For af/> as, i.e. for compres-
++Jr++ + + 0.9
++++
' 10
+ p=l/10+ '
20
+ '
'
0.9
30 40 50 number of deposited planes
Fig. 4. Evolution of the surface parameter adi ff deduced f r o m t h e m a x i m a of the diffracted intensity curves as a function of the n u m b e r of deposited planes and for different dislocation densities.
sive stresses in the film, the strain is obtained by replac-ing b by - b in eqns. (3) and (4) and also in eqn. (5) to obtain the displacement; then adifr(h) can be deduced from Fig. 4 using the symmetry about adiff/ a s = 1. It should be noted that for a large number of deposited planes adifr/a s-..+ aav/as or af/as when the epitaxial dislocations relax all the lattice mismatch between the substrate and the film.
NI Junqua, J. Grilhb / Measurements of misfit dislocation density
40
4. Contribution of the planes under the surface to the diffracted intensity In the preceding section we assumed that only the atoms at the free surface are diffracting. The validity of this hypothesis depends on the absorption length of the incident wave. In R H E E D experiments, the grazing incidence and detection angles mean that a long mean free path through the sample is associated with penetration normal to the surface of only a few atomic layers. We used the approximation that, for the fifth diffraction plane, the intensity becomes divided by 100, then the contribution of the j t h plane under the surface to the diffracted intensity is obtained with the attenuation factor exp( - j a / ~), ~ ~ a, and is negligible for j > 4. Calculations of the strain in planes under the free surface were performed. The stress component axx(x, z, h) of the dislocation in a semi-infinite solid is obtained by superposition of the stress fields of the dislocation and the image and with the supplementary term as shown by Head [5]. Summing the contributions of the dislocations on a row and using eqn. (4), we obtain the expression for the strain ex.,.(x, z, h) for dislocations at a distance xo. The analytic form of exx(x, z, h) is very large so only an example of exx(X) in the four planes near the free surface is shown in Fig. 5 when the density of epitaxial dislocations is p = 1/10 and the number of deposited planes is five. The displacement ux(x, z, h) on planes under the surface is calculated and summing eqn. (6) with the above attenuation coefficient we obtain the corresponding intensity curve I(Ak~). -0.7
i
a/a b/xo
a
i
9=1/20 p=l/lO
0.98
12 four diffracting planes 0.96
0.96 •
adi ff/a s
A
•
/~
0.94
0.94
0.92
?
0.92
: ~
•
0.9
• l
10
20
[]
•
[]
i
I
30 40 number of deposited planes
0.9 50
Fig. 6. E v o l u t i o n o f the p a r a m e t e r adi ff as a f u n c t i o n o f the n u m b e r o f d e p o s i t e d p l a n e s f o r d i s l o c a t i o n densities p = 1/10 a n d p = 1/20 in the f o l l o w i n g t w o cases: m , A , s u r f a c e p l a n e o n l y diffracting; ~ , L , f o u r d i f f r a c t i n g planes.
The variations of the parameter adiff for p = 1/10 and p = 1/20 are presented in Fig. 6 when only the surface plane is diffracting and when the contribution to the diffracted wave of four atomic planes under the surface is accounted for. The figure confirms that the contribution of the planes under the surface is weak for the R H E E D experimental conditions.
5. Conclusion
i
8
In epitaxial thin films with misfit dislocations, the surface strain presents strong variations for small film thicknesses (h << Xo). This inhomogeneity decreases as h increases and disappears for h ~ xo. One of the main consequences is that the surface parameter adi~ measured by diffraction methods is different from aav and varies with h for a constant dislocation density. The direct comparison of measured and computed intensity diffraction, or the comparison between the surface parameters ad~frdeduced from these curves (Fig. 4), gives a method of obtaining the real dislocation densities from such experiments. These corrections are needed only when h < Xo.
-- - surface .... plane 1 - - - plane 2 plane 3
-0.8 "
•
•
0.98
-0.9
-1.t
Acknowledgments -1.2 0
I 0.2
t 0.4
I 0.6
I 0.8 x/x
Fig. 5. P a r a m e t e r v a r i a t i o n s in p l a n e s u n d e r t h e free s u r f a c e f o r p = 1/10 a n d a film t h i c k n e s s h = 5a.
The authors would like to thank M. Piecuch and V. Pierron-Bohnes for helpful discussions and comments and M. Small for reading the manuscript and making useful suggestions.
N. Junqua, J. Grilhk / Measurements of misfit dislocation density
References 1 Ch. Chatillon and J. Massies, in FundamentalPhenomena in Nucleation and Growth of Crystals, Materials Science Forum, Vols. 59-60, Trans Tech, Aedermannsdorf, Switzerland, 1990, pp. 230-285. 2 Ph. Oudy, in Fundamental Phenomena in Nucleation and Growth of Crystals, Materials Science Forum, Vols. 59-60, Trans Tech, Aedermannsdorf, Switzerland, 1990, pp. 581-616. 3 W. D. Nix, Mechanical properties of thin films, Metall. Trans. A, 20 (1989) 2217-2245. 4 J. W. Matthews, J. Vac. Sci. Technol., 12 (1975) 126. 5 J. W. Matthews, A. E. Blakeslee and S. Mader, Thin Solid Films, 33 (1976) 253. 6 A. K. Head, Philos. Mag., 44 (1953) 92.
41
7 J. P. Hirth and J. Lothe, Theory and Dislocations, Wiley, New York, 1982. 8 P. M. Morse and H. Feshback, Method~ of Theoretical Physics, Vol. 1, McGraw-Hill, New York, 1953, p. 413. 9 S. Timoshenko and J. N. Goodier, Theory of Elasticity, McGrawHill, New York, 1951, p. 32. 10 A. Guinier, Thborie et Technique de la Radiocristallographie, 3rd edn., Dunod, Paris, 1964. 11 M. Piecuch and L. Nevot, in Fundamental Phenomena in Nucleation and Growth of Crystals, Materials Science Forum, Vols. 59-60, Trans Tech, Aedermannsdorf, Switzerland, 1990, pp. 93139. 12 H. Hfith, Surfaces and Interfaces of Solids, 2nd edn., Springer, New York, 1993, pp. 201-209.