- Email: [email protected]
Stress, strain and elastic energy at nanometric Ge dots on Si(0 0 1)
Applied Surface Science 188 (2002) 4±8
Stress, strain and elastic energy at nanometric Ge dots on Si(0 0 1) P. Raiteria,*, F. Valentinottib, L. Miglioa a
INFM and Department of Materials Science, University of Milano-Bicocca, V. Cozzi 53, 20125 Milan, Italy b Quadrics Supercomputers World Ltd., V. Marcellina 11, 00131 Rome, Italy
Abstract We perform molecular dynamics simulations to obtain the stress and strain distributions for Ge pyramids with {1 0 5} facets on Si(0 0 1). We show that the strain induced in the substrate is large and increasing with the pyramid size: up to 0.7% for the 22 nm in base, and corresponds to substrate bending below the pyramid. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Silicon; Germanium; Epitaxy; Dots; Stress; Strain
1. Introduction In lattice mismatched heteroepitaxy, the strain distribution plays an important role in determining the morphology of the ®lm. In the case of Ge growth on Si(0 0 1) (4.2% mismatch) the growth follows the Stranski±Krastanov mode, i.e. after the formation of a thin wetting layer (WL) as thick as three monolayers (ML), some pyramids with {1 0 5} facets appear ({1 0 5} Ge pyramids) [1]. The nucleation of three-dimensional (3D) islands is usually explained by the energy gain obtained by the strain relief in Ge dots. An accurate measurement of strain with spatial resolution is quite complex and few experimental techniques (RHEED, XRD and Fourier transform TEM) can only provide average or qualitative estimations [2±4]. Stress estimation has been recently achieved by atomic force microscopy and LEED investigation [5]. From the simulative point of view, ®nite elements (FE) and elastic continuum calcula*
Corresponding author. Tel.: 39-02-6448-5212; fax: 39-02-6448-5400. E-mail address: [email protected] (P. Raiteri).
tions provide a good accuracy for strain estimates at distances far form the dot, while they fail close to the interfaces [6]. In this work, we show a direct estimation of stress and strain in {1 0 5} Ge pyramids, by means of molecular dynamics relaxations, the (computational details are described elsewhere [7]). Much attention is paid to the calculation of the elastic ®eld in the silicon substrate induced by the dot and by the 2 8 reconstruction of the WL surface. Comparison to a coherent layer by layer (FM) Ge ®lm is also reported. The atomic interactions are computed according to the Tersoff potential [8], which allows for a good description of the elastic properties of Si±Ge compounds. Our model is composed by a 43 nm wide and 8.1 nm thick Si substrate, covered by a Ge wetting layer (3ML) plus pyramids with {1 0 5} faces. We simulated pyramids with base side as large as 5, 11, 16 and 22 nm. Periodic boundary conditions are set along the x and y directions and the two bottom Si planes are ®xed at their 0 K bulk positions, the temperature at which stress and strain are estimated. Starting from the strained systems, the con®guration at equilibrium is achieved by a simulated annealing
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 2 - 4
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
at 600 K, ®nally quenched at 0 K. The stress tensor at the atomic scale is computed according to [9]. 1 X a b sab i r f rijb fija (1) 4O j ij ij where a; b x; y; z, O the atomic volume, rij the distance of atom i from atom j and fij is the force that atom j exerts on atom i. Therefore, a negative value of sab means compressive stress. The strain tensor at the atomic scale can be computed as the transformation matrix acting on the positions of the nearest neighbors, from the undeformed bulk positions to the distorted ones [10]. A qualitative representation of the stress in the {1 0 5} pyramids is provided with a suitable gray scale in (0 1 0) cross-sections while quantitative plots of the strain are displayed along h0 0 1i rods passing through the pyramid apex or the square edges of its base. 2. Results In Figs. 1 and 2, the sxx and szz components of the stress tensor are shown in (0 1 0) cross-sections. Here
5
we do not report any picture relative to the smaller {1 0 5} pyramid, as it does not provide any further information. They indicate an overall expansion of the Si substrate (tensile stress), just below the pyramid and that the deformation gets deeper and deeper with the pyramid size. Due to the absence of lateral constraints, the stress relaxation in the topmost part of the pyramids (gray area) is larger and increases with the pyramid size. In Fig. 3, the shear stress component sxz is shown. Here the odd symmetry of the map (white on the right side and black on the left side) indicates the substrate bending provided by the central part of the pyramid pulling upward the substrate, which is an expected feature of such systems [2,9,11]. For sake of comparison, the vertical stress component szz computed in a (0 1 0) cross-section of a 2 8 reconstructed coherent FM Ge ®lm (4ML thick) over the same Si substrate is reported in Fig. 4. It is clear that no complex stress distribution is obtained when the dot is not present, but for the periodic modulation in the subsurface region, as induced by the Ge 2 8 reconstruction. A more quantitative analysis of the sxz component in the topmost layer 2 8 reconstructed, indicates
Fig. 1. In-plane stress sxx for Ge part of the system (a) and for the Si substrate underneath (b). Topside down we show the results for the 11, 16 and 22 nm wide {1 0 5} pyramids, whereas the thickness of the Si layer is 8.1 nm.
6
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
Fig. 2. The same as Fig. 1 for the vertical szz component.
values of 0.5±1.5 GPa, in good agreement to the measurements by Wedler et al. [5]. The behavior of the strain in the Si substrate with the penetration depth can be obtained from Figs. 5 and 6. We show the ezz
component (Fig. 5) and the hydrostatic deformation exx eyy ezz (Fig. 6), computed along rods passing through the pyramid apex (top) or its base edges (bottom). The Si/Ge interface is placed at 8.1 nm.
Fig. 3. The same as Fig. 1 for the shear sxz component.
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
7
Fig. 4. The same as Fig. 2 for a coherent Ge ®lm.
Here we report also the data relative to the smaller {1 0 5} pyramid. (5 nm). In this case, the deformation goes to zero at the bottom of the Si substrate. Just below the {1 0 5} pyramids (top panel) a positive maximum for ezz, gets larger and shifts deeper and deeper in the substrate, as the volume of the pyramid increases. Therefore, despite the deformation under the larger {1 0 5} pyramids still displays a relevant
value at the bottom of our ®nite system, we believe our system to be a realistic description also for thicker substrates. In Fig. 6, we note that the strain just below the pyramid edge (bottom panels) is always negative (i.e. compressive), much larger than the one corresponding to the vertical line passing by the pyramid apex. This feature is produced by a large compression, barely visible in the stress maps, induced by the
Fig. 5. Starting by the bottom of the Si substrate we report the ezz component of the strain computed along h0 0 1i rods passing through the pyramid apex (top panel) and the base edge (bottom panel).
8
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
Fig. 6. The same as in Fig. 5 for the hydrostatic strain.
volume expansion below the central part of the pyramid. 3. Conclusions We believe that the large strain induced in the substrate by the {1 0 5} pyramids plays an important role both in the energetics of the dot and in possible Si band gap modulations. Investigations on this latter issue are in progress. Acknowledgements The authors would like to acknowledge H. von KaÈnel and A. Rastelli (ETH Zurich) for helpful discussion.
References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11]
A. Vailionis, et al., Phys. Rev. Lett. 85 (2000) 3672. A.A. Williams, et al., Phys. Rev. B 43 (1991) 5001. A.I. Nikiforov, et al., Thin Solid Films 380 (2000) 158. T. Ide, A. Sakai, K. Shimizu, Thin Solid Films 357 (1999) 22. G. Wedler, J. Walz, T. Hesjedal, E. Chilla, R. Koch, Phys. Rev. Lett. 80 (1998) 2382. Y. Kikuchi, H. Sugii, K. Shintani, J. Appl. Phys. 89 (2001) 1191. P. Raiteri, F. Valentinotti, L. Miglio, V.E. Borisenko, S.V. Gaponenko, V.S. Gurin (Eds.), Physics, Chemistry and Application of Nanostructures, World Scienti®c, Singapore, 2001, pp. 69±75. J. Tersoff, Phys. Rev. B 39 (1989) 5566. W. Yu, A. Madhukar, Phys. Rev. Lett. 79 (1997) 905. C. Pryor, J. Kim, W. Wang, A.J. Williamson, A. Zunger, J. Appl. Phys. 83 (1998) 2548. D.J. Eaglesham, M. Cerullo, Phys. Rev. Lett. 64 (1990) 1943.
Stress, strain and elastic energy at nanometric Ge dots on Si(0 0 1) P. Raiteria,*, F. Valentinottib, L. Miglioa a
INFM and Department of Materials Science, University of Milano-Bicocca, V. Cozzi 53, 20125 Milan, Italy b Quadrics Supercomputers World Ltd., V. Marcellina 11, 00131 Rome, Italy
Abstract We perform molecular dynamics simulations to obtain the stress and strain distributions for Ge pyramids with {1 0 5} facets on Si(0 0 1). We show that the strain induced in the substrate is large and increasing with the pyramid size: up to 0.7% for the 22 nm in base, and corresponds to substrate bending below the pyramid. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Silicon; Germanium; Epitaxy; Dots; Stress; Strain
1. Introduction In lattice mismatched heteroepitaxy, the strain distribution plays an important role in determining the morphology of the ®lm. In the case of Ge growth on Si(0 0 1) (4.2% mismatch) the growth follows the Stranski±Krastanov mode, i.e. after the formation of a thin wetting layer (WL) as thick as three monolayers (ML), some pyramids with {1 0 5} facets appear ({1 0 5} Ge pyramids) [1]. The nucleation of three-dimensional (3D) islands is usually explained by the energy gain obtained by the strain relief in Ge dots. An accurate measurement of strain with spatial resolution is quite complex and few experimental techniques (RHEED, XRD and Fourier transform TEM) can only provide average or qualitative estimations [2±4]. Stress estimation has been recently achieved by atomic force microscopy and LEED investigation [5]. From the simulative point of view, ®nite elements (FE) and elastic continuum calcula*
Corresponding author. Tel.: 39-02-6448-5212; fax: 39-02-6448-5400. E-mail address: [email protected] (P. Raiteri).
tions provide a good accuracy for strain estimates at distances far form the dot, while they fail close to the interfaces [6]. In this work, we show a direct estimation of stress and strain in {1 0 5} Ge pyramids, by means of molecular dynamics relaxations, the (computational details are described elsewhere [7]). Much attention is paid to the calculation of the elastic ®eld in the silicon substrate induced by the dot and by the 2 8 reconstruction of the WL surface. Comparison to a coherent layer by layer (FM) Ge ®lm is also reported. The atomic interactions are computed according to the Tersoff potential [8], which allows for a good description of the elastic properties of Si±Ge compounds. Our model is composed by a 43 nm wide and 8.1 nm thick Si substrate, covered by a Ge wetting layer (3ML) plus pyramids with {1 0 5} faces. We simulated pyramids with base side as large as 5, 11, 16 and 22 nm. Periodic boundary conditions are set along the x and y directions and the two bottom Si planes are ®xed at their 0 K bulk positions, the temperature at which stress and strain are estimated. Starting from the strained systems, the con®guration at equilibrium is achieved by a simulated annealing
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 2 - 4
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
at 600 K, ®nally quenched at 0 K. The stress tensor at the atomic scale is computed according to [9]. 1 X a b sab i r f rijb fija (1) 4O j ij ij where a; b x; y; z, O the atomic volume, rij the distance of atom i from atom j and fij is the force that atom j exerts on atom i. Therefore, a negative value of sab means compressive stress. The strain tensor at the atomic scale can be computed as the transformation matrix acting on the positions of the nearest neighbors, from the undeformed bulk positions to the distorted ones [10]. A qualitative representation of the stress in the {1 0 5} pyramids is provided with a suitable gray scale in (0 1 0) cross-sections while quantitative plots of the strain are displayed along h0 0 1i rods passing through the pyramid apex or the square edges of its base. 2. Results In Figs. 1 and 2, the sxx and szz components of the stress tensor are shown in (0 1 0) cross-sections. Here
5
we do not report any picture relative to the smaller {1 0 5} pyramid, as it does not provide any further information. They indicate an overall expansion of the Si substrate (tensile stress), just below the pyramid and that the deformation gets deeper and deeper with the pyramid size. Due to the absence of lateral constraints, the stress relaxation in the topmost part of the pyramids (gray area) is larger and increases with the pyramid size. In Fig. 3, the shear stress component sxz is shown. Here the odd symmetry of the map (white on the right side and black on the left side) indicates the substrate bending provided by the central part of the pyramid pulling upward the substrate, which is an expected feature of such systems [2,9,11]. For sake of comparison, the vertical stress component szz computed in a (0 1 0) cross-section of a 2 8 reconstructed coherent FM Ge ®lm (4ML thick) over the same Si substrate is reported in Fig. 4. It is clear that no complex stress distribution is obtained when the dot is not present, but for the periodic modulation in the subsurface region, as induced by the Ge 2 8 reconstruction. A more quantitative analysis of the sxz component in the topmost layer 2 8 reconstructed, indicates
Fig. 1. In-plane stress sxx for Ge part of the system (a) and for the Si substrate underneath (b). Topside down we show the results for the 11, 16 and 22 nm wide {1 0 5} pyramids, whereas the thickness of the Si layer is 8.1 nm.
6
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
Fig. 2. The same as Fig. 1 for the vertical szz component.
values of 0.5±1.5 GPa, in good agreement to the measurements by Wedler et al. [5]. The behavior of the strain in the Si substrate with the penetration depth can be obtained from Figs. 5 and 6. We show the ezz
component (Fig. 5) and the hydrostatic deformation exx eyy ezz (Fig. 6), computed along rods passing through the pyramid apex (top) or its base edges (bottom). The Si/Ge interface is placed at 8.1 nm.
Fig. 3. The same as Fig. 1 for the shear sxz component.
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
7
Fig. 4. The same as Fig. 2 for a coherent Ge ®lm.
Here we report also the data relative to the smaller {1 0 5} pyramid. (5 nm). In this case, the deformation goes to zero at the bottom of the Si substrate. Just below the {1 0 5} pyramids (top panel) a positive maximum for ezz, gets larger and shifts deeper and deeper in the substrate, as the volume of the pyramid increases. Therefore, despite the deformation under the larger {1 0 5} pyramids still displays a relevant
value at the bottom of our ®nite system, we believe our system to be a realistic description also for thicker substrates. In Fig. 6, we note that the strain just below the pyramid edge (bottom panels) is always negative (i.e. compressive), much larger than the one corresponding to the vertical line passing by the pyramid apex. This feature is produced by a large compression, barely visible in the stress maps, induced by the
Fig. 5. Starting by the bottom of the Si substrate we report the ezz component of the strain computed along h0 0 1i rods passing through the pyramid apex (top panel) and the base edge (bottom panel).
8
P. Raiteri et al. / Applied Surface Science 188 (2002) 4±8
Fig. 6. The same as in Fig. 5 for the hydrostatic strain.
volume expansion below the central part of the pyramid. 3. Conclusions We believe that the large strain induced in the substrate by the {1 0 5} pyramids plays an important role both in the energetics of the dot and in possible Si band gap modulations. Investigations on this latter issue are in progress. Acknowledgements The authors would like to acknowledge H. von KaÈnel and A. Rastelli (ETH Zurich) for helpful discussion.
References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11]
A. Vailionis, et al., Phys. Rev. Lett. 85 (2000) 3672. A.A. Williams, et al., Phys. Rev. B 43 (1991) 5001. A.I. Nikiforov, et al., Thin Solid Films 380 (2000) 158. T. Ide, A. Sakai, K. Shimizu, Thin Solid Films 357 (1999) 22. G. Wedler, J. Walz, T. Hesjedal, E. Chilla, R. Koch, Phys. Rev. Lett. 80 (1998) 2382. Y. Kikuchi, H. Sugii, K. Shintani, J. Appl. Phys. 89 (2001) 1191. P. Raiteri, F. Valentinotti, L. Miglio, V.E. Borisenko, S.V. Gaponenko, V.S. Gurin (Eds.), Physics, Chemistry and Application of Nanostructures, World Scienti®c, Singapore, 2001, pp. 69±75. J. Tersoff, Phys. Rev. B 39 (1989) 5566. W. Yu, A. Madhukar, Phys. Rev. Lett. 79 (1997) 905. C. Pryor, J. Kim, W. Wang, A.J. Williamson, A. Zunger, J. Appl. Phys. 83 (1998) 2548. D.J. Eaglesham, M. Cerullo, Phys. Rev. Lett. 64 (1990) 1943.