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Role of surface step on misfit dislocation nucleation and critical thickness in semiconductor heterostructures
MATERIALS SCIENCE & ENGINEERING ELSEVIER
B
Materials Science and Engineering B31 (1995) 299-303
Role of surface step on misfit dislocation nucleation and critical thickness in semiconductor heterostructures M. Ichimura 1, J. Narayan Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7916, USA Received 10 September 1994
Abstract
We discuss the role of surface step formation on misfit dislocation nucleation and critical thickness in thin film semiconductor heterostructures. On the basis of an atomistic model, it is shown that the energy change due to the step formation is negative or positive depending upon the sign of the misfit. The step formation energy can even be negative for compressive misfit stress in the heterolayer, while it is definitely positive for tensile misfit stress. This conclusion is in contrast to the classical model where the step energy is always positive and independent of the sign of the misfit. The step formation energy influences the critical thickness and the energy barrier for dislocation nucleation. Using a simple atomistic simulation, we show that the critical thickness depends upon the sign of the misfit; for example, it changes from 4 nm for Ge films on Si(100) substrates to 6 nm for Si films on Ge(100) substrates having the same misfit.
Keywords: Semiconductor heterostructures; Step formation; Dislocation nucleation; Critical thickness
1. Introduction
Understanding of nucleation of misfit dislocations in lattice-mismatched heterostructures is the key to controlling and reducing dislocations in these heterostructures before they can be used for advanced semic o n d u c t o r devices. In d i a m o n d or zincblende s e m i c o n d u c t o r heterostructures with a (100) interface, misfit dislocations are mainly introduced by glide on {111} planes, which can result f r o m the bowing of threading dislocations or the nucleation of half-loops f r o m the surface [1]. In b o t h cases, a step is f o r m e d at the surface by the creation of a misfit dislocation if the surface is initially flat. In s o m e analyses of the misfit dislocation nucleation, the energy of the step is included on the basis of the m a c r o s c o p i c classical theory [1-4]. Fig. 1 shows the m a c r o s c o p i c picture of step formation. T h e step f o r m a t i o n leads to an increase of surface area, and the tension (the energy of the step per unit length) is given by
E = 7b sin 0
(1)
~On leave from Department of Electrical and Computer Engineering, Nagoya Institute of Technology, Nagoya 466, Japan. Elsevier Science S.A. SSDI 0921-5107(94)01146-X
where b is the magnitude of Burgers vector, 0 the angle between the dislocation line and the Burgers vector, and 7 the surface energy per unit area of the step surface [1,5]. Since y is positive, the macroscopic theory predicts that the surface energy is necessarily increased by the dislocation nucleation on the initially flat surface. This energy increase leads to an increase in the critical thickness. If the nucleation of dislocation is considered at a step on the surface, there is a nucleation barrier if the step size is increased or modified but the barrier decreases or disappears if the step is m e n d e d [5].
surface
,s •J
glide plane ,•
(a)
]~ b sine
(b) Fig. 1. Classical picture of the surface step formation: (a) flat surface; (b) surface after the dislocation nucleation.
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M. lehimura, J. Narayan
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In this paper, we discuss the step formation associated with the misfit dislocation nucleation based on an atomistic model. From a simple geometrical consideration, we conclude that the classical treatment of the step in covalently b o n d e d heretostructures is not adequate.
2. Atomistic model of the step formation We consider a (100) heterostructure composed of two diamond cubic or zincblende structure materials. T h e arrangement of atoms at the surface is shown in Fig. 2(a), where we view the crystal along the [011] direction. It should be mentioned that the semiconductor surfaces are usually reconstructed. T h e reconstruction is not considered initially but its effect is taken into account later on. T h e dashed line in Fig. 2(a) represents the glide plane of a 60 ° a/2{110) dislocation. W h e n the lefthand side of the plane glides upward and a 60 ° dislocation with a / 2 ~ 1 1 0 ) Burgers vector is introduced, the surface step is formed as shown in Fig. 2(b), which is equivalent to removal of a plane. T h e macroscopic picture of these surfaces is the same as in Fig. 1. Thus, the macroscopic theory predicts an increase in surface energy due to the step formation. T h e surface energy of
glide plane
(a)
(b)
(c) Fig. 2. Atom arrangement of the unreconstructed surfaces (a) before and (b), (c) after the dislocation nucleation. The surface orientation is (100), and the crystal is viewed along {011). The left-hand side of the glide plane glides upward in (b) and downward in (c).
a covalent crystal is mainly due to the presence of dangling bonds on the surface. Thus, in order to roughly estimate the surface energy atomistically, we count the number of dangling bonds before and after the step formation. Before the step formation, the number of dangling bonds is ten as depicted in the figure. After the step formation, it reduced to nine. Therefore, the surface energy is predicted to decrease rather than increase because of the step formation. (We consider here a small cell with five surface atoms, but the conclusion is the same for any size of cell. T h e formation of one dislocation line deletes one row of dangling bonds.) Of course, this is not a general conclusion because the number of dangling bonds depends on the surface structure. However, there is at least a possibility that the surface energy is reduced by the step formation, and such a possibility cannot be deduced from the classical picture. Next, we consider a glide in the opposite direction, which is equivalent to introduction of an extra half plane in the film. T h e surface atom arrangement after the glide is shown in Fig. 2(c). T h e Burgers vector of the dislocation is the same for both cases (Fig. 2(b) and (c)), and thus the surface energy is the same according to the macroscopic theory. However, if we count the number of dangling bonds, the surface energy of Fig. 2(c) is not thought to be the same as in Fig. 2(b). T h e number of the dangling bonds in Fig. 2(c) is 11, whereas there are ten and nine in Fig. 2(a) and 2(b), respectively. To discuss the difference between Fig. 2(b) and (c), we should take into account the fact that the surface energy is a function of not only the surface area and the orientation, but also of the interatomic distances on the surface. For example, if the interatomic spacing increases without a change in the surface area, the areal density of surface atoms and dangling bonds inevitably decreases and so does the surface energy. In a bulk material, the atomic spacing at the surface can be assumed to be constant under a constant external force. However, in the heterostructure, the atomic spacing can be changed by introduction of a dislocation. In Fig. 2(b), the interatomic spacing is larger than that in Fig. 2(a). T h e glide shown in Fig. 2(b) occurs in structures where the lattice constant of the heterolayer is larger than that of the substrate, such as Ge, GaAs or AlAs films on Si. Thus, the misfit dislocation relaxes the compressive strain and increases the atomic spacing in the layer. Therefore, the number of surface atoms or dangling bonds can be decreased even though the surface area increases by the step surface area. However, the atomic spacing becomes smaller owing to the dislocation in the case of Fig. 2(c). This type of glide occurs in layers with smaller lattice constant than that of the substrate, such as Si on Ge, GaAs or AlAs
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substrate. In this case, the number of surface dangling bonds becomes larger than in Fig. 2(a) and (b). It should be noted that the layer shown in Fig. 2(c) always has one extra row of surface atoms compared with that in Fig. 2(b); this conclusion holds for any reconstructed surfaces. T h e layer in Fig. 2(c) has one extra half (011) atomic plane owing to the misfit dislocation, while the substrate has one extra half plane (the layer has one missing half plane) in Fig. 2(b). Thus the layer in Fig. 2(c) has two more atomic planes, and two (011) atomic planes have one row of surface atoms. In other words, since the displacement of the surface atom along the surface due to the edge component of a 60 ° misfit dislocation is just half of the unit vector of the two-dimensional surface lattice, there is a difference of one surface unit between the two cases of the opposite glides. However, we cannot say that there is always a difference in the number of dangling bonds between the flat surface and the surface after the nucleation of one dislocation, because the displacement due to one dislocation is only half of the surface unit vector. In the classical picture, the dislocations are preferentially introduced at steps so that the step is eliminated by the glide. Even when we start with a surface with steps, we reach the same conclusion as described above. We consider the removal of a bi-atomic layer at the surface in the left-hand side of the glide plane in Fig. l(a). After the glide shown in Fig. l(b), this surface becomes a flat surface, and the number of dangling bonds decreases from nine to eight, i.e. it decreases by one as in the case of the initially fiat surface. However, if we remove a bi-atomic layer from the right-hand side of the glide plane in Fig. l(a) and consider the glide shown in Fig. l(c), the number of dangling bonds increases from nine to ten. In this case, the surface energy will increase even though the step is eliminated by the glide. This is in contrast to the results of the classical theory. In the above discussion, we neglect the surface reconstruction. In an actual (100) surface, the surface atoms form a dimer. In addition, the stable structure of a double step is predicted to be different from those in Fig. 2(b) and (c) [6]. Thus we discuss how the surface reconstruction influences the above results, taking a (100) surface of Group IV semiconductors as an example. To reach a stable double step, we begin with a surface arrangement shown in Fig. 3(a). In this figure, we view a (2× 1) reconstructed (100) surface along [011]. The dimer bonds on the surface are perpendicular to the sheet and parallel to the dislocation line. While each surface atom has two dangling bonds on an unreconstructed surface as shown in Fig. 2, it has only one dangling bond in Fig. 3. Therefore, the number of dangling bonds is five in the figure.
g)jde plane
(a)
(b')
(b)
A
(c') (e)
(c") Fig. 3. (2 x 1) reconstructed surfaces (a) before and (b)-(c") after the dislocation nucleation. The dimerized bonds of the topmost atoms are perpendicular to the sheet. After the glide, the atoms are expected to be rearranged to form a stable double step as shown in (b'), (c') or (c"). Fig. 3(b) shows the surface after the glide, as in Fig. 2(b). The atom labeled A is at a very unstable position in Fig. 3(b), but it can rebond to the atoms of the lower layer to form the stable double step structure [6] (Fig. 3(b')). Then, the number of dangling bonds becomes four, smaller by one than in Fig. 3(a). The surface structure after the glide in the opposite direction is shown in Fig. 3(c). In this case, we cannot make the stable double step structure because of the difference in the number of atoms. The stable double step is formed if atom A is removed or an atom is added to site B. Therefore, atom A will move to site B of another step site, and half of the step becomes Fig. 3 (c') and the other half Fig. 3(c"). In both the cases, the number of dangling bonds is five, the same as in Fig. 3(a) and greater by one than in Fig. 3(b).
3. Effects on the critical thickness
The actual step energy will depend on the detailed surface structure, which in turn depends on various growth conditions. However, from the above argument,
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we can conclude that there is a tendency that the dislocation nucleation from a flat surface will be easier under compressive stress; the step formation energy can even be negative on certain surfaces if the dislocation relaxes the compressive stress, while it is definitely positive if the dislocation relaxes the tensile stress. Therefore, the critical thickness and the nucleation barrier will be different between these two cases. To show how large the difference is, we give here results of our atomistic calculation of the critical thickness for Ge/Si, where the heterolayer (Ge) has a lattice constant 4% larger than that of the substrate (Si). We consider a two-dimensional atom cell with a width of 5 nm, which corresponds to the spacing between 60 ° dislocations in a completely relaxed Ge/Si structure. T h e cell has no dislocation (coherent state) or one 60 ° dislocation at the interface (relaxed state). T h e energy of the cell is calculated under periodic boundary conditions using Stillinger-Weber potentials [7,8], and every atom is sequentillay moved in the direction of the force to minimize the cell energy. T h e growth of Ge on Si(100) can be two-dimensional (2-D) or 3-D, depending on growth conditions, particularly the temperature. We have shown, in our recent work, that the growth m o d e switches from 2-D to 3-D at 300 °C in Ge films on Si(100) produced by pulsed laser deposition [9]. Fig. 4 shows the energy per unit area after the minimization for (100) Ge/Si heterostructures as a function of the Ge thickness. T h e open circles represent the energy o f coherent Ge layers, while the dark triangles show energy of relaxed structures where the misfit stress in one direction is completely relaxed by an array of 60 ° dislocations. As shown in the figure, these two energies cross at a thickness of 28 monoatomic layers 10
Ge/Si 8
5
x x
•
6 x x
4
•
•
• O
O x
~J
o
2
•
o
o
coherent
•
relaxed
0
o
I
o
10 Ge
I
20 thickness
i
I
30
I
(ML), or 4 nm. T h e value of critical thickness obtained by the present atomistic calculation is larger than that of the classical theory [1], primarily because the classical theory ignores the core energy contribution and the nature of the dislocation. (Details of this are to be published shortly.) In calculating the energy of the relaxed state, we adopted the simple surface model shown in Fig. 2, i.e. the surface energy is thought to be decreased owing to the dislocation nucleation. If Ge is grown on a hypothetical substrate which has a lattice constant 4% larger than that of Ge (or Si is grown on Ge), the dislocation creates surface dangling bonds as shown in Fig. 2(c). In this case, we must add the energy due to these dangling bonds (1 eV nm 2) to the energy of the relaxed state. T h e n the relaxed state has a higher energy even at a thickness of 35 M L (5 nm), as shown by the crosses on the figure. It should be pointed out that this situation will not alter if there is already a step on the surface. In the case of an initially flat surface, we have a transition from ten to nine and from ten to eleven dangling bonds corresponding to Fig. 2(b) and (c), respectively. In the presence of a step, the corresponding transition will be from nine to eight and from nine to ten, respectively. Therefore, the critical thickness can differ greatly, depending on the sign of the mismatch, in thin film heterostructures.
4. Conclusion We have discussed the role of surface steps on misfit dislocation nucleation in thin film semiconductor heterostructures on the basis of an atomistic model. T h e energy change due to the step formation is negative or positive depending upon the sign of the misfit. T h e surface energy can even decrease for compressive misfit stress in the film, while it must increase for tensile misfit stress. T h e same conclusion holds in the case of step elimination: for a film under tensile misfit stress, the surface energy will increase even when the step is mended by the dislocation. T h o s e results are in contrast to the classical model where the surface energy is always increased (decreased) by the step creation (elimination), irrespective of the sign of the misfit.
Acknowledgments
40
(ML)
Fig. 4. Energies of (100)Ge/Si structures as a function of Ge thickness: o, coherently strained Ge layers; A, relaxed structures where the misfit stress of one direction is relaxed by 60 ° dislocations. If the sign of the mismatch is opposite, the energy of the relaxed state would become higher by 1 eV mm -, as shown by the crosses (x) because of the increase in surface dangling bond density (unreconstructed surfaces are assumed).
We would like to thank Drs. K. Jagannadham and S. Okytyabrsky for useful discussions.
References [1] J.W. Matthews, J. Vac. Sci. Technol., 12 (1975) 126. [2] S.V. Kamat and J.P. Hirth, J. Appl. Phys., 67(1990)6844.
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[3] R. Hull and J.C. Bean, J. Vac. Sci. Technol. A, 7(1989) 2580. [4] K. Jagannadham and J. Narayan, Mater. Sci. Eng., B8 ( 1991 ) 107. [5] J.P. Hirth, in The Relation between the Structure and Mechanical Properties of Alloys, HM Stationery Office, London, 1963, p. 217.
[6] [7] [8] [9]
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D.J. Chadi, Phys. Rev. Lett., 59(1987) 1691. EH. Stillinger and T.A. Weber, Phys. Rev. B, 31 (1985) 5262. K. Ding and H.C. Andersen, Phys. Rev. B, 34 (1986) 6987. S. Oktyabrsky, H. Wu, R.D. Vispute and J. Narayan, Phil Mag. A , in press.
B
Materials Science and Engineering B31 (1995) 299-303
Role of surface step on misfit dislocation nucleation and critical thickness in semiconductor heterostructures M. Ichimura 1, J. Narayan Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7916, USA Received 10 September 1994
Abstract
We discuss the role of surface step formation on misfit dislocation nucleation and critical thickness in thin film semiconductor heterostructures. On the basis of an atomistic model, it is shown that the energy change due to the step formation is negative or positive depending upon the sign of the misfit. The step formation energy can even be negative for compressive misfit stress in the heterolayer, while it is definitely positive for tensile misfit stress. This conclusion is in contrast to the classical model where the step energy is always positive and independent of the sign of the misfit. The step formation energy influences the critical thickness and the energy barrier for dislocation nucleation. Using a simple atomistic simulation, we show that the critical thickness depends upon the sign of the misfit; for example, it changes from 4 nm for Ge films on Si(100) substrates to 6 nm for Si films on Ge(100) substrates having the same misfit.
Keywords: Semiconductor heterostructures; Step formation; Dislocation nucleation; Critical thickness
1. Introduction
Understanding of nucleation of misfit dislocations in lattice-mismatched heterostructures is the key to controlling and reducing dislocations in these heterostructures before they can be used for advanced semic o n d u c t o r devices. In d i a m o n d or zincblende s e m i c o n d u c t o r heterostructures with a (100) interface, misfit dislocations are mainly introduced by glide on {111} planes, which can result f r o m the bowing of threading dislocations or the nucleation of half-loops f r o m the surface [1]. In b o t h cases, a step is f o r m e d at the surface by the creation of a misfit dislocation if the surface is initially flat. In s o m e analyses of the misfit dislocation nucleation, the energy of the step is included on the basis of the m a c r o s c o p i c classical theory [1-4]. Fig. 1 shows the m a c r o s c o p i c picture of step formation. T h e step f o r m a t i o n leads to an increase of surface area, and the tension (the energy of the step per unit length) is given by
E = 7b sin 0
(1)
~On leave from Department of Electrical and Computer Engineering, Nagoya Institute of Technology, Nagoya 466, Japan. Elsevier Science S.A. SSDI 0921-5107(94)01146-X
where b is the magnitude of Burgers vector, 0 the angle between the dislocation line and the Burgers vector, and 7 the surface energy per unit area of the step surface [1,5]. Since y is positive, the macroscopic theory predicts that the surface energy is necessarily increased by the dislocation nucleation on the initially flat surface. This energy increase leads to an increase in the critical thickness. If the nucleation of dislocation is considered at a step on the surface, there is a nucleation barrier if the step size is increased or modified but the barrier decreases or disappears if the step is m e n d e d [5].
surface
,s •J
glide plane ,•
(a)
]~ b sine
(b) Fig. 1. Classical picture of the surface step formation: (a) flat surface; (b) surface after the dislocation nucleation.
300
M. lehimura, J. Narayan
/
Materials Science and Engineering B31 (1995) 299-303
In this paper, we discuss the step formation associated with the misfit dislocation nucleation based on an atomistic model. From a simple geometrical consideration, we conclude that the classical treatment of the step in covalently b o n d e d heretostructures is not adequate.
2. Atomistic model of the step formation We consider a (100) heterostructure composed of two diamond cubic or zincblende structure materials. T h e arrangement of atoms at the surface is shown in Fig. 2(a), where we view the crystal along the [011] direction. It should be mentioned that the semiconductor surfaces are usually reconstructed. T h e reconstruction is not considered initially but its effect is taken into account later on. T h e dashed line in Fig. 2(a) represents the glide plane of a 60 ° a/2{110) dislocation. W h e n the lefthand side of the plane glides upward and a 60 ° dislocation with a / 2 ~ 1 1 0 ) Burgers vector is introduced, the surface step is formed as shown in Fig. 2(b), which is equivalent to removal of a plane. T h e macroscopic picture of these surfaces is the same as in Fig. 1. Thus, the macroscopic theory predicts an increase in surface energy due to the step formation. T h e surface energy of
glide plane
(a)
(b)
(c) Fig. 2. Atom arrangement of the unreconstructed surfaces (a) before and (b), (c) after the dislocation nucleation. The surface orientation is (100), and the crystal is viewed along {011). The left-hand side of the glide plane glides upward in (b) and downward in (c).
a covalent crystal is mainly due to the presence of dangling bonds on the surface. Thus, in order to roughly estimate the surface energy atomistically, we count the number of dangling bonds before and after the step formation. Before the step formation, the number of dangling bonds is ten as depicted in the figure. After the step formation, it reduced to nine. Therefore, the surface energy is predicted to decrease rather than increase because of the step formation. (We consider here a small cell with five surface atoms, but the conclusion is the same for any size of cell. T h e formation of one dislocation line deletes one row of dangling bonds.) Of course, this is not a general conclusion because the number of dangling bonds depends on the surface structure. However, there is at least a possibility that the surface energy is reduced by the step formation, and such a possibility cannot be deduced from the classical picture. Next, we consider a glide in the opposite direction, which is equivalent to introduction of an extra half plane in the film. T h e surface atom arrangement after the glide is shown in Fig. 2(c). T h e Burgers vector of the dislocation is the same for both cases (Fig. 2(b) and (c)), and thus the surface energy is the same according to the macroscopic theory. However, if we count the number of dangling bonds, the surface energy of Fig. 2(c) is not thought to be the same as in Fig. 2(b). T h e number of the dangling bonds in Fig. 2(c) is 11, whereas there are ten and nine in Fig. 2(a) and 2(b), respectively. To discuss the difference between Fig. 2(b) and (c), we should take into account the fact that the surface energy is a function of not only the surface area and the orientation, but also of the interatomic distances on the surface. For example, if the interatomic spacing increases without a change in the surface area, the areal density of surface atoms and dangling bonds inevitably decreases and so does the surface energy. In a bulk material, the atomic spacing at the surface can be assumed to be constant under a constant external force. However, in the heterostructure, the atomic spacing can be changed by introduction of a dislocation. In Fig. 2(b), the interatomic spacing is larger than that in Fig. 2(a). T h e glide shown in Fig. 2(b) occurs in structures where the lattice constant of the heterolayer is larger than that of the substrate, such as Ge, GaAs or AlAs films on Si. Thus, the misfit dislocation relaxes the compressive strain and increases the atomic spacing in the layer. Therefore, the number of surface atoms or dangling bonds can be decreased even though the surface area increases by the step surface area. However, the atomic spacing becomes smaller owing to the dislocation in the case of Fig. 2(c). This type of glide occurs in layers with smaller lattice constant than that of the substrate, such as Si on Ge, GaAs or AlAs
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substrate. In this case, the number of surface dangling bonds becomes larger than in Fig. 2(a) and (b). It should be noted that the layer shown in Fig. 2(c) always has one extra row of surface atoms compared with that in Fig. 2(b); this conclusion holds for any reconstructed surfaces. T h e layer in Fig. 2(c) has one extra half (011) atomic plane owing to the misfit dislocation, while the substrate has one extra half plane (the layer has one missing half plane) in Fig. 2(b). Thus the layer in Fig. 2(c) has two more atomic planes, and two (011) atomic planes have one row of surface atoms. In other words, since the displacement of the surface atom along the surface due to the edge component of a 60 ° misfit dislocation is just half of the unit vector of the two-dimensional surface lattice, there is a difference of one surface unit between the two cases of the opposite glides. However, we cannot say that there is always a difference in the number of dangling bonds between the flat surface and the surface after the nucleation of one dislocation, because the displacement due to one dislocation is only half of the surface unit vector. In the classical picture, the dislocations are preferentially introduced at steps so that the step is eliminated by the glide. Even when we start with a surface with steps, we reach the same conclusion as described above. We consider the removal of a bi-atomic layer at the surface in the left-hand side of the glide plane in Fig. l(a). After the glide shown in Fig. l(b), this surface becomes a flat surface, and the number of dangling bonds decreases from nine to eight, i.e. it decreases by one as in the case of the initially fiat surface. However, if we remove a bi-atomic layer from the right-hand side of the glide plane in Fig. l(a) and consider the glide shown in Fig. l(c), the number of dangling bonds increases from nine to ten. In this case, the surface energy will increase even though the step is eliminated by the glide. This is in contrast to the results of the classical theory. In the above discussion, we neglect the surface reconstruction. In an actual (100) surface, the surface atoms form a dimer. In addition, the stable structure of a double step is predicted to be different from those in Fig. 2(b) and (c) [6]. Thus we discuss how the surface reconstruction influences the above results, taking a (100) surface of Group IV semiconductors as an example. To reach a stable double step, we begin with a surface arrangement shown in Fig. 3(a). In this figure, we view a (2× 1) reconstructed (100) surface along [011]. The dimer bonds on the surface are perpendicular to the sheet and parallel to the dislocation line. While each surface atom has two dangling bonds on an unreconstructed surface as shown in Fig. 2, it has only one dangling bond in Fig. 3. Therefore, the number of dangling bonds is five in the figure.
g)jde plane
(a)
(b')
(b)
A
(c') (e)
(c") Fig. 3. (2 x 1) reconstructed surfaces (a) before and (b)-(c") after the dislocation nucleation. The dimerized bonds of the topmost atoms are perpendicular to the sheet. After the glide, the atoms are expected to be rearranged to form a stable double step as shown in (b'), (c') or (c"). Fig. 3(b) shows the surface after the glide, as in Fig. 2(b). The atom labeled A is at a very unstable position in Fig. 3(b), but it can rebond to the atoms of the lower layer to form the stable double step structure [6] (Fig. 3(b')). Then, the number of dangling bonds becomes four, smaller by one than in Fig. 3(a). The surface structure after the glide in the opposite direction is shown in Fig. 3(c). In this case, we cannot make the stable double step structure because of the difference in the number of atoms. The stable double step is formed if atom A is removed or an atom is added to site B. Therefore, atom A will move to site B of another step site, and half of the step becomes Fig. 3 (c') and the other half Fig. 3(c"). In both the cases, the number of dangling bonds is five, the same as in Fig. 3(a) and greater by one than in Fig. 3(b).
3. Effects on the critical thickness
The actual step energy will depend on the detailed surface structure, which in turn depends on various growth conditions. However, from the above argument,
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we can conclude that there is a tendency that the dislocation nucleation from a flat surface will be easier under compressive stress; the step formation energy can even be negative on certain surfaces if the dislocation relaxes the compressive stress, while it is definitely positive if the dislocation relaxes the tensile stress. Therefore, the critical thickness and the nucleation barrier will be different between these two cases. To show how large the difference is, we give here results of our atomistic calculation of the critical thickness for Ge/Si, where the heterolayer (Ge) has a lattice constant 4% larger than that of the substrate (Si). We consider a two-dimensional atom cell with a width of 5 nm, which corresponds to the spacing between 60 ° dislocations in a completely relaxed Ge/Si structure. T h e cell has no dislocation (coherent state) or one 60 ° dislocation at the interface (relaxed state). T h e energy of the cell is calculated under periodic boundary conditions using Stillinger-Weber potentials [7,8], and every atom is sequentillay moved in the direction of the force to minimize the cell energy. T h e growth of Ge on Si(100) can be two-dimensional (2-D) or 3-D, depending on growth conditions, particularly the temperature. We have shown, in our recent work, that the growth m o d e switches from 2-D to 3-D at 300 °C in Ge films on Si(100) produced by pulsed laser deposition [9]. Fig. 4 shows the energy per unit area after the minimization for (100) Ge/Si heterostructures as a function of the Ge thickness. T h e open circles represent the energy o f coherent Ge layers, while the dark triangles show energy of relaxed structures where the misfit stress in one direction is completely relaxed by an array of 60 ° dislocations. As shown in the figure, these two energies cross at a thickness of 28 monoatomic layers 10
Ge/Si 8
5
x x
•
6 x x
4
•
•
• O
O x
~J
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I
(ML), or 4 nm. T h e value of critical thickness obtained by the present atomistic calculation is larger than that of the classical theory [1], primarily because the classical theory ignores the core energy contribution and the nature of the dislocation. (Details of this are to be published shortly.) In calculating the energy of the relaxed state, we adopted the simple surface model shown in Fig. 2, i.e. the surface energy is thought to be decreased owing to the dislocation nucleation. If Ge is grown on a hypothetical substrate which has a lattice constant 4% larger than that of Ge (or Si is grown on Ge), the dislocation creates surface dangling bonds as shown in Fig. 2(c). In this case, we must add the energy due to these dangling bonds (1 eV nm 2) to the energy of the relaxed state. T h e n the relaxed state has a higher energy even at a thickness of 35 M L (5 nm), as shown by the crosses on the figure. It should be pointed out that this situation will not alter if there is already a step on the surface. In the case of an initially flat surface, we have a transition from ten to nine and from ten to eleven dangling bonds corresponding to Fig. 2(b) and (c), respectively. In the presence of a step, the corresponding transition will be from nine to eight and from nine to ten, respectively. Therefore, the critical thickness can differ greatly, depending on the sign of the mismatch, in thin film heterostructures.
4. Conclusion We have discussed the role of surface steps on misfit dislocation nucleation in thin film semiconductor heterostructures on the basis of an atomistic model. T h e energy change due to the step formation is negative or positive depending upon the sign of the misfit. T h e surface energy can even decrease for compressive misfit stress in the film, while it must increase for tensile misfit stress. T h e same conclusion holds in the case of step elimination: for a film under tensile misfit stress, the surface energy will increase even when the step is mended by the dislocation. T h o s e results are in contrast to the classical model where the surface energy is always increased (decreased) by the step creation (elimination), irrespective of the sign of the misfit.
Acknowledgments
40
(ML)
Fig. 4. Energies of (100)Ge/Si structures as a function of Ge thickness: o, coherently strained Ge layers; A, relaxed structures where the misfit stress of one direction is relaxed by 60 ° dislocations. If the sign of the mismatch is opposite, the energy of the relaxed state would become higher by 1 eV mm -, as shown by the crosses (x) because of the increase in surface dangling bond density (unreconstructed surfaces are assumed).
We would like to thank Drs. K. Jagannadham and S. Okytyabrsky for useful discussions.
References [1] J.W. Matthews, J. Vac. Sci. Technol., 12 (1975) 126. [2] S.V. Kamat and J.P. Hirth, J. Appl. Phys., 67(1990)6844.
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[3] R. Hull and J.C. Bean, J. Vac. Sci. Technol. A, 7(1989) 2580. [4] K. Jagannadham and J. Narayan, Mater. Sci. Eng., B8 ( 1991 ) 107. [5] J.P. Hirth, in The Relation between the Structure and Mechanical Properties of Alloys, HM Stationery Office, London, 1963, p. 217.
[6] [7] [8] [9]
303
D.J. Chadi, Phys. Rev. Lett., 59(1987) 1691. EH. Stillinger and T.A. Weber, Phys. Rev. B, 31 (1985) 5262. K. Ding and H.C. Andersen, Phys. Rev. B, 34 (1986) 6987. S. Oktyabrsky, H. Wu, R.D. Vispute and J. Narayan, Phil Mag. A , in press.