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Misfit dislocations: experimental evidence for nucleation of partial dislocations in thin films of gold on Pd(001)
Thin Solid Films, 87(1982) L7-L12
L7
LeRer Misfit dislocations: experimental evidence for nucleation of partial dislocations in thin films of gold on Pd(O01) F. HILA AND M. GILLET Laboratoire de Microscopie et Diffractions Electroniques associd au CNRS, E.R.A.545, Facuttd des Sciences et Techniques de Saint-Jdr6me, 13397 Marseitte Cddex 13 (France.) (Received June 30, 1981; accepted December 10, 1981)
1. Introduction When a metallic deposit is epitaxially grown on a monocrystalline substrate with the same crystallographic structure but with a slightly different lattice constant, interfacial atoms must adjust their positions to accommodate the lattice misfit. A theoretical treatment of this problem was first given by Frank and van der Merwe 1. This theory, which assumes a minimum interfacial energy, is concerned with the way in which the overgrowth accommodates the natural misfit: as the deposition begins, the misfit is entirely accommodated by elastic strain; as the deposit thickness increases, interfacial dislocations are progressively introduced to minimize the elastic strain energy. Epitaxial growth has been investigated for a wide range of bicrystal systems and generally the experiments are qualitatively in agreement with van der Merwe's predictions. However, the exact characteristics of interfacial dislocations are not always known. One of the most interesting problems is the nucleation of partial misfit dislocations during coherence loss. Two mechanisms have been invoked for the nucleation of interfacial dislocations 2. The first is the glide of dislocations extending from the substrate through the overgrowth. The second is the nucleation of half-loops at the deposit-free surface. In these two mechanisms misfit dislocations can be either perfect or partial. Partial misfit dislocations of the Shockley type have been observed on the bicrystals Co/Cu(001) 3.4, Co/Ni(111) s, Ag/Au(111) 6 and Ag/Pd(001) 7. It is possible to determine the Burgers vectors by electron microscopy techniques, but in practice this investigation is difficult owing to the high density of complex contrasts in the electron micrographs. In 1974, Cherns s showed, by a careful examination of short interfacial dislocations segments in Pd/Au(001), that they were for the most part of the Shockley type. In this paper we shall describe direct observations of the stacking faults associated with partial misfit dislocations in Au/Pd(001) thin films. These observations are in good agreement with the minimum interfacial energy calculations. 2. Specimen preparation Epitaxial bilayers were obtained by evaporation of gold onto thin palladium substrates which had previously been epitaxially grown onto air-cleaved NaC1 0040-6090/82/0000-0000/$02.75
© ElsevierSequoia/Printedin The Netherlands
L8
LETTERS
crystals held at 250 °C. The thickness of the substrates was about 350 ~. Before the deposition they were thermally cleaned in an ultrahigh vacuum at 300 °C for 3 h; then they were cooled to 100 °C during deposition of the gold. The evaporation of the gold was performed from a Knudsen cell in a vacuum of 10-a Torr with an evaporation rate of about 1/~ s- 1. The thickness of the deposit was measured with a quartz monitor; it varied in the range 1-50 A. 3. Observations The Au/Pd(001) bilayers were examined by electron diffraction and electron microscopy. As the thickness increases, the electron diffraction pattern and the contrast in the electron micrographs change. During the initial deposition of the gold no contrast changes are observed and no additional spots appear on the diffraction patterns. When the deposit thickness is greater than 10/~ we observe a high density of contrast features which appear as short lines in the two (110) directions. Their density increases with the deposit thickness. These lines are interpreted as interfacial dislocation images. When the deposit thickness exceeds 20 ~, these line contrasts are often blurred by moir6 images, but by tilting the specimen it is possible to find suitable conditions for their observation. Figures l(a), l(b) and l(c) are a bright field and two dark field images recorded with g = 020 and g = 200 reflections respectively Electron diffraction patterns for deposit thicknesses hAu exceeding 10/~ exhibit streaks of the type expected from stacking faults 9. These satellite streaks are seen when the specimen is tilted around a (100) axis, as shown on Fig. l(d). Figures l(e) and l(f) are dark field images obtained with the M x and M 2 reflections respectively of Fig. l(d). They exhibit a high density of contrast features corresponding to the line contrasts observed on bright field micrographs. 4. Interpretation From the electron diffraction patterns, which contain streaks arising from deposit reflections and not from reflections from the palladium substrate, and by examination of dark field micrographs it is possible to interpret the line contrasts as stacking faults located in the gold deposit and limited by partial misfit dislocations in the interface plane. It should be noted that gold and palladium are miscible in all proportions; this means that the observed contrasts cannot be attributed to platelets of precipitates resulting from alloying. Partial misfit dislocations and stacking faults will be generated if the energy of the bicrystal containing these defects at the equilibrium concentration is smaller than the energy of a similar bicrystal containing an equilibrium concentration of perfect dislocations. Imperfect dislocations can be either of the Frank type or the Shockley type. In imperfect Frank-type dislocations the (111) plane is destroyed and this is consistent with the fact that the deposit parameter becomes larger than that corresponding to the pseudomorphic stage. Hence in the following we consider imperfect dislocations to be of the Frank type. We consider two kinds of dislocations which can nucleate from the free bicrystal surface: a perfect edge dislocation in the [110] direction made by climb in a (110) plane with a Burgers' vector b = 21-110]
LETTERS
L9
L
(a)
(b)
(c)
(d)
(e) ~
(f)
L\~
Fig. 1. A bicrystal of gold on a Pd(001) substrate: (a)-(c) transmission electron micrographs showing interfacial dislocations ((a) bright field image; (b) dark field image withg = 200; (c) dark field image with g = 020); (d) electron diffraction pattern exhibiting satellite streaks M 1 and M 2 produced by stacking faults in the deposit; (e), (f) dark field images obtained with reflections M1 and M2 respectively.
L10
LETTERS
and an imperfect dislocation of the Frank type climbing in a (1 lI) plane with a Burgers' vector b = 3[11i] We calculate the equilibrium energy of the bicrystal according to the deposit thickness assuming that the misfit is accommodated either by elastic strain or by elastic strain and misfit dislocations. The energy of a bicrystal which contains N interfacial dislocations is given by E T = Edisl -at- Edh q- E Mq- E F
In this expression Edisl is the edge dislocation energy per sample area and is approximately equal to 7 N 4•(1 - v~ In
+1
where Gi is the shear modulus of the interface with 2
1
1
Gi -- ~ss~ G d
Gs and Gd being shear moduli of the substrate and the deposit respectively (Gd = 2.75x 1011 d y n c m - 2 ; Gs = 4.41 x 1011 d y n c m -2 (ref. 10)) and v is Poisson's ratio (0.4). Edh is the elastic strain energy for a deposit of thickness h and is given by 1--V
Edh = Gd l~-~vhfd
2
where fd is the part of the misfit accommodated by elastic strain. If the misfit is entirely accommodated by elastic strain, fd is equal to the natural misfit f (4.8~). EF is the stacking fault energy per unit film area and is given by h E F = Nydco s
where Ydis the specific stacking fault energy (55 erg cm- 2 (ref. 11)) and ~bis the angle between the normal to the film surface and the stacking fault plane. EM is the energy of the surface steps produced by the formation of the stacking faults. This energy is related to the change E in the number of broken bonds for atoms which assume edge positions along the steps. EM is given by n'/£
Eu = Nad nL
where n' is the number of atoms in edge positions per unit length of the step, n is the number of atoms per unit area of the (001) plane, L is the number of broken bonds for atoms in the (001) surface plane and ad is the specific surface energy (2271 erg cm-2 (ref. 12)). If the misfit dislocation is perfect Eu = EF = 0. Figure 2 shows interfacial configurations for a perfect dislocation (Fig. 2(a)) and for a Frank partial dislocation (Fig. 2(b)). In both cases the dislocation spacing D is given by the
LETTERS
Li 1
coincidence A1,B 1 and
A2,B2,
so
that
d,i'd, D = - dd' --
ds
where d, and d d' a r e the spacings of the substrate and the deposit atoms (the substrate is assumed to be unstrained): ds = as t/2 and dd' = ad '~/2 where as and a d' are the substrate and the strained deposit lattice parameters respectively. The deposit elastic strain is given by fd
ad
--
adr Gd --
dg !
!
I
=
q-,~k+-' -4+. I I
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.
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~-~-4
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Fig. 2. Interface configurations for an Au/Pd(001) bicrysta| which contains (a) a perfect interracial dislocation or (b) a partial dislocation of the Frank type. Fig. 3. Energy of an Au/Pd(001) bicrystal: curve a, interface with an equilibrium concentration of perfect misfit dislocations; curve b, interface with an equilibrium concentration of partial misfit dislocations and stacking faults in the deposit; curve c, natural misfit entirely accommodated by elastic strain.
L 12
LETTERS
In Fig. 3 curves a and b represent the energy of a bicrystal which contains an equilibrium concentration of misfit dislocations, according to the condition dE-r d a d'
=0
for an interface with perfect dislocations (E x = Edisl+ Edh) (curve a) and for an interface with Frank partial dislocations (Ex = Edisl + Edh -'1-EM + EF) (curve b). The straight line c corresponds to the case of a misfit entirely accommodated by elastic strain. For the deposit thickness range 0-5/~, curves a and b coincide with the straight line c. This means that in the initial stages of deposition the deposit is elastically strained by the substrate. For deposits thicker than 5/~, interfacial dislocations can be nucleated to minimize elastic strain. The energy corresponding to the bicrystal Au/Pd(001) with partial misfit dislocations and stacking faults is lower than that of a similar specimen which contains only perfect dislocations. Hence in the deposit thickness range 0-50/~, which was studied in our experiments, partial misfit dislocations will be favoured, and our observations are in agreement with this conclusion. 1 2 3 4 5 6 7
F . C . Frank and J. H. van der Merwe, Proc. R. Soc. London, Ser. A, 198 (1949) 216. J.W. Matthews, Phys. Thin Films, 4 (1967) 137. W . A . Jesser and J. W. Matthews, Philos. Mag., 17 (1968) 461. W . A . Jesser, J. Appl. Phys., 41 (1) (1969) 39. W . A . Jesser and J. W. Matthews, Acta Metall., 16 (1968) 1307. H . C . Snyman and J. A. Engelbrecht, Acta Metall., 21 (1973) 479. J.W. Matthews, J. Appl. Phys., 42 (13) (1971) 5640.
8 D. ChernsandM. J. Stowell, ThinSolidFilms, 29(1)(1975)107. 9 M.J. Whelan and P. B. Hirsh, Philos. Mag., 2 (1957) 1303. 10 Handbook of Chemistry and Physics, Chemical Rubber Co., Cleveland, OH, 41st edn., p. 2143. 11 J. Price Hirth and J. Lothe, Theory of Dislocations, p. 764. 12 H. Suzuki and C. S. Barret, Acta Metall., 6 (1958) 156.
L7
LeRer Misfit dislocations: experimental evidence for nucleation of partial dislocations in thin films of gold on Pd(O01) F. HILA AND M. GILLET Laboratoire de Microscopie et Diffractions Electroniques associd au CNRS, E.R.A.545, Facuttd des Sciences et Techniques de Saint-Jdr6me, 13397 Marseitte Cddex 13 (France.) (Received June 30, 1981; accepted December 10, 1981)
1. Introduction When a metallic deposit is epitaxially grown on a monocrystalline substrate with the same crystallographic structure but with a slightly different lattice constant, interfacial atoms must adjust their positions to accommodate the lattice misfit. A theoretical treatment of this problem was first given by Frank and van der Merwe 1. This theory, which assumes a minimum interfacial energy, is concerned with the way in which the overgrowth accommodates the natural misfit: as the deposition begins, the misfit is entirely accommodated by elastic strain; as the deposit thickness increases, interfacial dislocations are progressively introduced to minimize the elastic strain energy. Epitaxial growth has been investigated for a wide range of bicrystal systems and generally the experiments are qualitatively in agreement with van der Merwe's predictions. However, the exact characteristics of interfacial dislocations are not always known. One of the most interesting problems is the nucleation of partial misfit dislocations during coherence loss. Two mechanisms have been invoked for the nucleation of interfacial dislocations 2. The first is the glide of dislocations extending from the substrate through the overgrowth. The second is the nucleation of half-loops at the deposit-free surface. In these two mechanisms misfit dislocations can be either perfect or partial. Partial misfit dislocations of the Shockley type have been observed on the bicrystals Co/Cu(001) 3.4, Co/Ni(111) s, Ag/Au(111) 6 and Ag/Pd(001) 7. It is possible to determine the Burgers vectors by electron microscopy techniques, but in practice this investigation is difficult owing to the high density of complex contrasts in the electron micrographs. In 1974, Cherns s showed, by a careful examination of short interfacial dislocations segments in Pd/Au(001), that they were for the most part of the Shockley type. In this paper we shall describe direct observations of the stacking faults associated with partial misfit dislocations in Au/Pd(001) thin films. These observations are in good agreement with the minimum interfacial energy calculations. 2. Specimen preparation Epitaxial bilayers were obtained by evaporation of gold onto thin palladium substrates which had previously been epitaxially grown onto air-cleaved NaC1 0040-6090/82/0000-0000/$02.75
© ElsevierSequoia/Printedin The Netherlands
L8
LETTERS
crystals held at 250 °C. The thickness of the substrates was about 350 ~. Before the deposition they were thermally cleaned in an ultrahigh vacuum at 300 °C for 3 h; then they were cooled to 100 °C during deposition of the gold. The evaporation of the gold was performed from a Knudsen cell in a vacuum of 10-a Torr with an evaporation rate of about 1/~ s- 1. The thickness of the deposit was measured with a quartz monitor; it varied in the range 1-50 A. 3. Observations The Au/Pd(001) bilayers were examined by electron diffraction and electron microscopy. As the thickness increases, the electron diffraction pattern and the contrast in the electron micrographs change. During the initial deposition of the gold no contrast changes are observed and no additional spots appear on the diffraction patterns. When the deposit thickness is greater than 10/~ we observe a high density of contrast features which appear as short lines in the two (110) directions. Their density increases with the deposit thickness. These lines are interpreted as interfacial dislocation images. When the deposit thickness exceeds 20 ~, these line contrasts are often blurred by moir6 images, but by tilting the specimen it is possible to find suitable conditions for their observation. Figures l(a), l(b) and l(c) are a bright field and two dark field images recorded with g = 020 and g = 200 reflections respectively Electron diffraction patterns for deposit thicknesses hAu exceeding 10/~ exhibit streaks of the type expected from stacking faults 9. These satellite streaks are seen when the specimen is tilted around a (100) axis, as shown on Fig. l(d). Figures l(e) and l(f) are dark field images obtained with the M x and M 2 reflections respectively of Fig. l(d). They exhibit a high density of contrast features corresponding to the line contrasts observed on bright field micrographs. 4. Interpretation From the electron diffraction patterns, which contain streaks arising from deposit reflections and not from reflections from the palladium substrate, and by examination of dark field micrographs it is possible to interpret the line contrasts as stacking faults located in the gold deposit and limited by partial misfit dislocations in the interface plane. It should be noted that gold and palladium are miscible in all proportions; this means that the observed contrasts cannot be attributed to platelets of precipitates resulting from alloying. Partial misfit dislocations and stacking faults will be generated if the energy of the bicrystal containing these defects at the equilibrium concentration is smaller than the energy of a similar bicrystal containing an equilibrium concentration of perfect dislocations. Imperfect dislocations can be either of the Frank type or the Shockley type. In imperfect Frank-type dislocations the (111) plane is destroyed and this is consistent with the fact that the deposit parameter becomes larger than that corresponding to the pseudomorphic stage. Hence in the following we consider imperfect dislocations to be of the Frank type. We consider two kinds of dislocations which can nucleate from the free bicrystal surface: a perfect edge dislocation in the [110] direction made by climb in a (110) plane with a Burgers' vector b = 21-110]
LETTERS
L9
L
(a)
(b)
(c)
(d)
(e) ~
(f)
L\~
Fig. 1. A bicrystal of gold on a Pd(001) substrate: (a)-(c) transmission electron micrographs showing interfacial dislocations ((a) bright field image; (b) dark field image withg = 200; (c) dark field image with g = 020); (d) electron diffraction pattern exhibiting satellite streaks M 1 and M 2 produced by stacking faults in the deposit; (e), (f) dark field images obtained with reflections M1 and M2 respectively.
L10
LETTERS
and an imperfect dislocation of the Frank type climbing in a (1 lI) plane with a Burgers' vector b = 3[11i] We calculate the equilibrium energy of the bicrystal according to the deposit thickness assuming that the misfit is accommodated either by elastic strain or by elastic strain and misfit dislocations. The energy of a bicrystal which contains N interfacial dislocations is given by E T = Edisl -at- Edh q- E Mq- E F
In this expression Edisl is the edge dislocation energy per sample area and is approximately equal to 7 N 4•(1 - v~ In
+1
where Gi is the shear modulus of the interface with 2
1
1
Gi -- ~ss~ G d
Gs and Gd being shear moduli of the substrate and the deposit respectively (Gd = 2.75x 1011 d y n c m - 2 ; Gs = 4.41 x 1011 d y n c m -2 (ref. 10)) and v is Poisson's ratio (0.4). Edh is the elastic strain energy for a deposit of thickness h and is given by 1--V
Edh = Gd l~-~vhfd
2
where fd is the part of the misfit accommodated by elastic strain. If the misfit is entirely accommodated by elastic strain, fd is equal to the natural misfit f (4.8~). EF is the stacking fault energy per unit film area and is given by h E F = Nydco s
where Ydis the specific stacking fault energy (55 erg cm- 2 (ref. 11)) and ~bis the angle between the normal to the film surface and the stacking fault plane. EM is the energy of the surface steps produced by the formation of the stacking faults. This energy is related to the change E in the number of broken bonds for atoms which assume edge positions along the steps. EM is given by n'/£
Eu = Nad nL
where n' is the number of atoms in edge positions per unit length of the step, n is the number of atoms per unit area of the (001) plane, L is the number of broken bonds for atoms in the (001) surface plane and ad is the specific surface energy (2271 erg cm-2 (ref. 12)). If the misfit dislocation is perfect Eu = EF = 0. Figure 2 shows interfacial configurations for a perfect dislocation (Fig. 2(a)) and for a Frank partial dislocation (Fig. 2(b)). In both cases the dislocation spacing D is given by the
LETTERS
Li 1
coincidence A1,B 1 and
A2,B2,
so
that
d,i'd, D = - dd' --
ds
where d, and d d' a r e the spacings of the substrate and the deposit atoms (the substrate is assumed to be unstrained): ds = as t/2 and dd' = ad '~/2 where as and a d' are the substrate and the strained deposit lattice parameters respectively. The deposit elastic strain is given by fd
ad
--
adr Gd --
dg !
!
I
=
q-,~k+-' -4+. I I
_,,_~
I I
I I
_..¢__~
I; °.~__ I
I L I I
I I
,
I
I ---
)--
I;
I I
1
I I
.J
---L-
-" ~9"" I
t,,: !1 ? - ~ , - , : ,--~--
.
.
.
.
!
~
.
Jl'"
!
i--;--4--~ --,~-~ I I
Deposit
1 1
i
--4-, I I
Su~a¢e
,--~-
!
I I
,
---
!
!
I
J
-
deposit
.
L÷.,-_ 4-. /
"I"-" ~I " q " I' ~ I
I
I
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,
,I
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I
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i
,_.~--..+-~--~'
I
I
,411
(a)
P l a n (1113
A
:q
/ C /
B
I I
/
•
B
C
/
A
/
/
!/,' d ~ ' V i / i / i
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o
+;s! I
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..... (b)
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I
•
='
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U.I
d
=
i
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•
c 1i"o3
100 i
~-~-4
I I lO
hAuf~J
Fig. 2. Interface configurations for an Au/Pd(001) bicrysta| which contains (a) a perfect interracial dislocation or (b) a partial dislocation of the Frank type. Fig. 3. Energy of an Au/Pd(001) bicrystal: curve a, interface with an equilibrium concentration of perfect misfit dislocations; curve b, interface with an equilibrium concentration of partial misfit dislocations and stacking faults in the deposit; curve c, natural misfit entirely accommodated by elastic strain.
L 12
LETTERS
In Fig. 3 curves a and b represent the energy of a bicrystal which contains an equilibrium concentration of misfit dislocations, according to the condition dE-r d a d'
=0
for an interface with perfect dislocations (E x = Edisl+ Edh) (curve a) and for an interface with Frank partial dislocations (Ex = Edisl + Edh -'1-EM + EF) (curve b). The straight line c corresponds to the case of a misfit entirely accommodated by elastic strain. For the deposit thickness range 0-5/~, curves a and b coincide with the straight line c. This means that in the initial stages of deposition the deposit is elastically strained by the substrate. For deposits thicker than 5/~, interfacial dislocations can be nucleated to minimize elastic strain. The energy corresponding to the bicrystal Au/Pd(001) with partial misfit dislocations and stacking faults is lower than that of a similar specimen which contains only perfect dislocations. Hence in the deposit thickness range 0-50/~, which was studied in our experiments, partial misfit dislocations will be favoured, and our observations are in agreement with this conclusion. 1 2 3 4 5 6 7
F . C . Frank and J. H. van der Merwe, Proc. R. Soc. London, Ser. A, 198 (1949) 216. J.W. Matthews, Phys. Thin Films, 4 (1967) 137. W . A . Jesser and J. W. Matthews, Philos. Mag., 17 (1968) 461. W . A . Jesser, J. Appl. Phys., 41 (1) (1969) 39. W . A . Jesser and J. W. Matthews, Acta Metall., 16 (1968) 1307. H . C . Snyman and J. A. Engelbrecht, Acta Metall., 21 (1973) 479. J.W. Matthews, J. Appl. Phys., 42 (13) (1971) 5640.
8 D. ChernsandM. J. Stowell, ThinSolidFilms, 29(1)(1975)107. 9 M.J. Whelan and P. B. Hirsh, Philos. Mag., 2 (1957) 1303. 10 Handbook of Chemistry and Physics, Chemical Rubber Co., Cleveland, OH, 41st edn., p. 2143. 11 J. Price Hirth and J. Lothe, Theory of Dislocations, p. 764. 12 H. Suzuki and C. S. Barret, Acta Metall., 6 (1958) 156.