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Kinetic effects in heteroepitaxial growth
Applied Surface Science 188 (2002) 1±3
Kinetic effects in heteroepitaxial growth J. Tersoff* IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10599, USA
Abstract Epitaxial growth is controlled by an interplay between thermodynamic and kinetic driving forces. The role of kinetics is especially obvious in unstable systems such as growth of strained heteroepitaxial layers. In particular, for growth of alloy strained layers, kinetic effects can dramatically in¯uence morphology and alloy composition. Some recent progress in understanding heteroepitaxial growth of strained alloy layers is brie¯y reviewed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Epitaxy; Stress; Strain; Growth; Kinetics
1. Introduction In heteroepitaxial growth, there is generally some mis®t between the lattice spacing of the epilayer and the substrate, leading to stress in the epilayer. This results in a well-known morphological instability [1±3]. This instability has important consequences for technological applications, where planar layers are required. Thus, understanding and controlling the instability is a problem of longstanding interest. The stress-driven morphological instability has been studied extensively for single-component ®lms (for review, see [4]). A ripple (a small sinusoidal variation of the surface height) causes some reduction of elastic stress at the peaks, with an increase in stress at the troughs. The resulting chemical-potential gradient drives diffusion from troughs to peaks, causing the height modulation to increase exponentially with time. The surface energy of the ®lm tends to oppose this, but at long wavelengths the elastic effect is always dominant for non-facetted systems.
* Tel.: 1-914-945-3138; fax: 1-914-945-4506. E-mail address: [email protected] (J. Tersoff).
In most applications, the strained layer is an alloy. But only recently has the stress-induced instability been properly understood in the case of a strained alloy. Here I summarize the key elements of some recent work on this problem [5±8]. This discussion is restricted to the case of an unfacetted surface, as occurs at high temperature. During growth of an alloy, any evolution of the surface morphology leads to non-uniform composition. This occurs for two reasons. The two kinds of atoms have different size, so they experience different chemical potentials and thus difference driving forces for diffusion. In addition, there is always some difference in the mobility of the respective atoms, so their kinetic response to the thermodynamic driving forces is different. 2. Effect of atomic size difference The effect of atomic size difference was treated by Spencer et al. [5]. If the two kinds of atom have the same surface diffusivity, the effect of their size difference is always to make the strained layer even less stable. Some earlier work had reached different
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 7 0 0 - 0
2
J. Tersoff / Applied Surface Science 188 (2002) 1±3
Fig. 1. Rate s of instability (at most-unstable wavelength) vs. deposition rate (i.e. growth rate, in dimensionless units) for the case described in [5]. The perturbation grows with time as exp(st). Here Z is the fractional difference in atomic size (lattice constant) between the two components.
conclusions as discussed in [5]. The reason for this disagreement is apparently that earlier treatments tried to simplify the problem. Because there are two types of atoms, and the compositional and morphological degrees of freedom of the system are coupled, the problem must be treated as a 2 2 tensor version of the classic one-component problem, where the tensor indices refer to the two types of atoms. Fig. 1 shows the degree of instability, as a function of atomic size difference and growth rate, when the two kinds of atoms have the same surface diffusivity (for details, see [5]). The key point here is that, when the atoms have different size, there is an increase in the instability rate. The magnitude of this increase depends on the growth rate, and is greatest at intermediate growth rates. 3. Effect of atomic mobility difference In general, the two components of an alloy will have different surface diffusivity. Diffusion is a thermally activated process, and the two types of atoms will generally have different activation energies for diffusion. As a result, the two diffusivities may easily differ by orders of magnitude. Therefore, differences in surface diffusivity are expected to play a key role in the behavior of strained alloy layers.
This problem was treated recently by Spencer et al. [6,7]. An important limiting case was treated earlier by Venezuela and Tersoff [8]. The effects of atomic size difference and atomic mobility difference can either add or oppose each other. Consider the example of SiGe on a Si substrate. If the surface is slightly non-planar, with a small ripple, the compressive stress is slightly relaxed at the top of the ripple. This creates a thermodynamic driving force for diffusion of atoms toward the top of the ripple. The Ge atom experiences a stronger driving force for diffusion than the Si atom, because of its size difference relative to the substrate. Moreover, the higher Ge diffusivity lets it diffuse to the top more effectively. Thus both factors combine to give Ge enrichment of the alloy at the top of the ripple. This in turn leads to greater strain relaxation there. The net effect is to increase the instability rate. The formal mathematical description of this behavior is given in [6,7]. Now consider the case of SiGe grown on a Ge substrate. In this case, it is the Si that has the mis®t with respect to the substrate, so the Si experiences a greater thermodynamic driving force toward the top of the ripple. However, insofar as the large mobility difference is the dominant effect, the Ge can still diffuse more effectively to the strain-relaxed top of the ripple, leaving Si enrichment in the troughs. This enrichment in the more mis®tting component, the Si, causes extra strain relaxation in the troughs. If this effect is large enough, it drives diffusion to the troughs and reduces the instability. In some systems, depending on atomic sizes, diffusivities, and growth rate, this effect can actually suppress the instability completely. In this case, the system is thermodynamically unstable, in that an increase in the ripple amplitude would lower the free energy. Yet the system is dynamically stable; the ripple amplitude actually decreases with time, even though this raises the free energy. Thus the ®lm remains planar during growth, with potentially important consequences for semiconductor technology. The behavior is illustrated in Fig. 2. 4. Comparison with experiment There are relatively few experiments that give direct quantitative information about the instability of
J. Tersoff / Applied Surface Science 188 (2002) 1±3
Fig. 2. Range of unstable wavenumbers for a sinusoidal perturbation vs. growth rate (in dimensionless units for the case described in [6]) for different values of the ratio b of the atomic mobilities of the two alloy components. For suf®ciently large mobility difference, the instability is completely suppressed above some critical growth rate. This is the case for the four curves terminating at the origin; dashed curve is marginal case, i.e. smallest b for which growth is stable above some critical deposition rate.
strained layers. A few relevant experiments are discussed brie¯y here. There is a clear experimental evidence that a SiGe alloy is more unstable under compression than under tension [9]. This was determined by growing the same alloy on both Si and Ge substrates. There are many possible explanations for this behavior, as discussed previously [10]. But it is intriguing to note that this is precisely what would be expected from the combined effects of atomic size and mobility difference for the SiGe system. Recently, two groups measured the most-unstable wavelength for SiGe alloys on Si, for a range of alloy
3
compositions [11,12]. Both found a surprising result, that the wavelength scales inversely with the strain (i.e. with the Ge fraction). In contrast, theories for single-component systems had predicted wavelength scaling as the inverse square of the strain. This apparent discrepancy can be resolved by recognizing that this system is an alloy with a very large mobility different between the two species. In the limit that Ge diffusion is very much faster than Si, the wavelength is found to scale inversely as the square of the Ge±Si atomic size difference, but only inversely with the alloy mis®t (i.e. the Ge fraction) [13]. This surprising behavior illustrates the important difference between alloys and pure materials in this context. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.J. Asaro, W.A. Tiller, Metall. Trans. 3 (1972) 1789. M.A. Grinfeld, Sov. Phys. Dokl. 31 (1986) 831. D.J. Srolovitz, Acta Metall. 37 (1989) 621. H. Gao, J. Mech. Phys. Solids 42 (1994) 741. B.J. Spencer, P.W. Voorhees, J. Tersoff, Phys. Rev. Lett. 84 (2000) 2449. B.J. Spencer, P.W. Voorhees, J. Tersoff, Appl. Phys. Lett. 76 (2000) 3022. B.J. Spencer, P.W. Voorhees, J. Tersoff, Unpublished result. P. Venezuela, J. Tersoff, Phys. Rev. B 58 (1998) 10871. Y.-H. Xie, et al., Phys. Rev. Lett. 73 (1994) 3006. J. Tersoff, Phys. Rev. Lett. 74 (1995) 4962. P. Suttert, M.G. Lagally, Phys. Rev. Lett. 84 (2000) 4637. R.M. Tromp, et al., Phys. Rev. Lett. 84 (2000) 4641. J. Tersoff, Phys. Rev. Lett. 85 (2000) 2843.
Kinetic effects in heteroepitaxial growth J. Tersoff* IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10599, USA
Abstract Epitaxial growth is controlled by an interplay between thermodynamic and kinetic driving forces. The role of kinetics is especially obvious in unstable systems such as growth of strained heteroepitaxial layers. In particular, for growth of alloy strained layers, kinetic effects can dramatically in¯uence morphology and alloy composition. Some recent progress in understanding heteroepitaxial growth of strained alloy layers is brie¯y reviewed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Epitaxy; Stress; Strain; Growth; Kinetics
1. Introduction In heteroepitaxial growth, there is generally some mis®t between the lattice spacing of the epilayer and the substrate, leading to stress in the epilayer. This results in a well-known morphological instability [1±3]. This instability has important consequences for technological applications, where planar layers are required. Thus, understanding and controlling the instability is a problem of longstanding interest. The stress-driven morphological instability has been studied extensively for single-component ®lms (for review, see [4]). A ripple (a small sinusoidal variation of the surface height) causes some reduction of elastic stress at the peaks, with an increase in stress at the troughs. The resulting chemical-potential gradient drives diffusion from troughs to peaks, causing the height modulation to increase exponentially with time. The surface energy of the ®lm tends to oppose this, but at long wavelengths the elastic effect is always dominant for non-facetted systems.
* Tel.: 1-914-945-3138; fax: 1-914-945-4506. E-mail address: [email protected] (J. Tersoff).
In most applications, the strained layer is an alloy. But only recently has the stress-induced instability been properly understood in the case of a strained alloy. Here I summarize the key elements of some recent work on this problem [5±8]. This discussion is restricted to the case of an unfacetted surface, as occurs at high temperature. During growth of an alloy, any evolution of the surface morphology leads to non-uniform composition. This occurs for two reasons. The two kinds of atoms have different size, so they experience different chemical potentials and thus difference driving forces for diffusion. In addition, there is always some difference in the mobility of the respective atoms, so their kinetic response to the thermodynamic driving forces is different. 2. Effect of atomic size difference The effect of atomic size difference was treated by Spencer et al. [5]. If the two kinds of atom have the same surface diffusivity, the effect of their size difference is always to make the strained layer even less stable. Some earlier work had reached different
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 7 0 0 - 0
2
J. Tersoff / Applied Surface Science 188 (2002) 1±3
Fig. 1. Rate s of instability (at most-unstable wavelength) vs. deposition rate (i.e. growth rate, in dimensionless units) for the case described in [5]. The perturbation grows with time as exp(st). Here Z is the fractional difference in atomic size (lattice constant) between the two components.
conclusions as discussed in [5]. The reason for this disagreement is apparently that earlier treatments tried to simplify the problem. Because there are two types of atoms, and the compositional and morphological degrees of freedom of the system are coupled, the problem must be treated as a 2 2 tensor version of the classic one-component problem, where the tensor indices refer to the two types of atoms. Fig. 1 shows the degree of instability, as a function of atomic size difference and growth rate, when the two kinds of atoms have the same surface diffusivity (for details, see [5]). The key point here is that, when the atoms have different size, there is an increase in the instability rate. The magnitude of this increase depends on the growth rate, and is greatest at intermediate growth rates. 3. Effect of atomic mobility difference In general, the two components of an alloy will have different surface diffusivity. Diffusion is a thermally activated process, and the two types of atoms will generally have different activation energies for diffusion. As a result, the two diffusivities may easily differ by orders of magnitude. Therefore, differences in surface diffusivity are expected to play a key role in the behavior of strained alloy layers.
This problem was treated recently by Spencer et al. [6,7]. An important limiting case was treated earlier by Venezuela and Tersoff [8]. The effects of atomic size difference and atomic mobility difference can either add or oppose each other. Consider the example of SiGe on a Si substrate. If the surface is slightly non-planar, with a small ripple, the compressive stress is slightly relaxed at the top of the ripple. This creates a thermodynamic driving force for diffusion of atoms toward the top of the ripple. The Ge atom experiences a stronger driving force for diffusion than the Si atom, because of its size difference relative to the substrate. Moreover, the higher Ge diffusivity lets it diffuse to the top more effectively. Thus both factors combine to give Ge enrichment of the alloy at the top of the ripple. This in turn leads to greater strain relaxation there. The net effect is to increase the instability rate. The formal mathematical description of this behavior is given in [6,7]. Now consider the case of SiGe grown on a Ge substrate. In this case, it is the Si that has the mis®t with respect to the substrate, so the Si experiences a greater thermodynamic driving force toward the top of the ripple. However, insofar as the large mobility difference is the dominant effect, the Ge can still diffuse more effectively to the strain-relaxed top of the ripple, leaving Si enrichment in the troughs. This enrichment in the more mis®tting component, the Si, causes extra strain relaxation in the troughs. If this effect is large enough, it drives diffusion to the troughs and reduces the instability. In some systems, depending on atomic sizes, diffusivities, and growth rate, this effect can actually suppress the instability completely. In this case, the system is thermodynamically unstable, in that an increase in the ripple amplitude would lower the free energy. Yet the system is dynamically stable; the ripple amplitude actually decreases with time, even though this raises the free energy. Thus the ®lm remains planar during growth, with potentially important consequences for semiconductor technology. The behavior is illustrated in Fig. 2. 4. Comparison with experiment There are relatively few experiments that give direct quantitative information about the instability of
J. Tersoff / Applied Surface Science 188 (2002) 1±3
Fig. 2. Range of unstable wavenumbers for a sinusoidal perturbation vs. growth rate (in dimensionless units for the case described in [6]) for different values of the ratio b of the atomic mobilities of the two alloy components. For suf®ciently large mobility difference, the instability is completely suppressed above some critical growth rate. This is the case for the four curves terminating at the origin; dashed curve is marginal case, i.e. smallest b for which growth is stable above some critical deposition rate.
strained layers. A few relevant experiments are discussed brie¯y here. There is a clear experimental evidence that a SiGe alloy is more unstable under compression than under tension [9]. This was determined by growing the same alloy on both Si and Ge substrates. There are many possible explanations for this behavior, as discussed previously [10]. But it is intriguing to note that this is precisely what would be expected from the combined effects of atomic size and mobility difference for the SiGe system. Recently, two groups measured the most-unstable wavelength for SiGe alloys on Si, for a range of alloy
3
compositions [11,12]. Both found a surprising result, that the wavelength scales inversely with the strain (i.e. with the Ge fraction). In contrast, theories for single-component systems had predicted wavelength scaling as the inverse square of the strain. This apparent discrepancy can be resolved by recognizing that this system is an alloy with a very large mobility different between the two species. In the limit that Ge diffusion is very much faster than Si, the wavelength is found to scale inversely as the square of the Ge±Si atomic size difference, but only inversely with the alloy mis®t (i.e. the Ge fraction) [13]. This surprising behavior illustrates the important difference between alloys and pure materials in this context. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.J. Asaro, W.A. Tiller, Metall. Trans. 3 (1972) 1789. M.A. Grinfeld, Sov. Phys. Dokl. 31 (1986) 831. D.J. Srolovitz, Acta Metall. 37 (1989) 621. H. Gao, J. Mech. Phys. Solids 42 (1994) 741. B.J. Spencer, P.W. Voorhees, J. Tersoff, Phys. Rev. Lett. 84 (2000) 2449. B.J. Spencer, P.W. Voorhees, J. Tersoff, Appl. Phys. Lett. 76 (2000) 3022. B.J. Spencer, P.W. Voorhees, J. Tersoff, Unpublished result. P. Venezuela, J. Tersoff, Phys. Rev. B 58 (1998) 10871. Y.-H. Xie, et al., Phys. Rev. Lett. 73 (1994) 3006. J. Tersoff, Phys. Rev. Lett. 74 (1995) 4962. P. Suttert, M.G. Lagally, Phys. Rev. Lett. 84 (2000) 4637. R.M. Tromp, et al., Phys. Rev. Lett. 84 (2000) 4641. J. Tersoff, Phys. Rev. Lett. 85 (2000) 2843.