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[0 0 1] zone-axis bright-field diffraction contrast from coherent Ge(Si) islands on Si(0 0 1)
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Ultramicroscopy 98 (2004) 239–247
[0 0 1] zone-axis bright-field diffraction contrast from coherent Ge(Si) islands on Si(0 0 1) X.Z. Liaoa,b, J. Zoua,c,*, D.J.H. Cockayned, S. Matsumurae a
Australian Key Centre for Microscopy and Microanalysis, The University of Sydney, Sydney, NSW 2006, Australia b Division of Materials Science and Technology, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Division of Materials and Centre for Microscopy and Microanalysis, The University of Queensland, QLD 4072, Australia d Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK e Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Fukuoka 812-8581, Japan Received 14 April 2003; received in revised form 8 July 2003 Dedicated to Professor Fang-hua Li on the occasion of her 70th birthday
Abstract Coherent Ge(Si)/Si(0 0 1) quantum dot islands grown by solid source molecular beam epitaxy at a growth temperature of 700 C were investigated using transmission electron microscopy working at 300 kV. The [0 0 1] zone-axis bright-field diffraction contrast images of the islands show strong periodicity with the change of the TEM sample substrate thickness and the period is equal to the effective extinction distance of the transmitted beam. Simulated images based on finite element models of the displacement field and using multi-beam dynamical diffraction theory show a high degree of agreement. Studies for a range of electron energies show the power of the technique for investigating composition segregation in quantum dot islands. r 2003 Elsevier B.V. All rights reserved. PACS: 68.37.Lp; 81.07.Ta; 81.05.Cy Keywords: TEM characterization; Image simulation; Quantum dot
1. Introduction Coherent hetero-epitaxial grown semiconductor islands such as Ge(Si)/Si [1] and InGaAs/GaAs [2] are of great interest because of their potential applications as semiconductor quantum dots *Corresponding author. Centre for Microscopy and Microanalysis, The University of Queensland, Queensland, QLD 4072, Australia. Tel.: +61-7-336-53977; fax: +61-7-336-54422. E-mail address: [email protected] (J. Zou).
(QDs). The opto-electronic properties of QDs depend strongly on the structural parameters including the shape, size, composition and strain distribution of the QDs [3,4]. Considerable effort has been made to investigate these structural parameters, with transmission electron microscopy (TEM) diffraction contrast being frequently used. For example, Liao et al. [5] investigated the composition of InGaAs/GaAs(0 0 1) QDs using the [0 0 1] multi-beam bright-field diffraction contrast imaging technique; Liu et al. [6] measured the
0304-3991/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2003.08.017
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strain evolution of coherent Ge islands on Si(0 0 1) using a TEM technique based on two-beam darkfield diffraction contrast imaging; and Benabbas et al. [7] and Androussi et al. [8] studied the shape and size of InAs/GaAs QDs by applying two-beam dynamical electron diffraction contrast imaging techniques. TEM diffraction contrast has been used for the investigation of coherent precipitates since the early 1960s when Ashby and Brown [9,10] showed that diffraction contrast from inclusions could be used to yield information about the magnitude of the strain. Image simulation is an important tool for comparing different strain models with experimental images and obtaining a reliable strain field distribution from a model structure is the important first step. Some early simulations of displacement fields of coherent precipitates were carried out within the framework of isotropic elasticity theory [9–11] while later simulations used anisotropic elasticity [12–15]. These early investigations calculated the strain fields using analytical expressions, but this is only possible for a few simple regular shapes such as spheres [13] and cubes [12]. However, with the increasing interest in Stranski-Krastanow [16] grown semiconductor QDs (which can be regarded as coherent precipitates), obtaining strain field distributions for more complicated shapes such as lenses or pyramids, and for unburied ‘‘precipitates’’, has become increasingly important. The improving speed of numerical computations has made it possible to calculate the displacement field for strained systems such as QDs using numerical methods rather than analytical expressions. For example, Christiansen et al. [17] used finite element analysis to obtain the displacement field of Ge(Si) islands grown on Si. Their study showed that the displacement field obtained in this way correctly explains features of convergent beam diffraction patterns. Scheerschmidt et al. [18] used molecular dynamics to derive the displacement field for CdSe/ZnSe QDs, and applied the result to the simulation of TEM images. In our earlier investigation, we have successfully employed finite element analysis to generate the strain field of unburied Ge(Si)/Si(0 0 1) [19] QD
islands and used the multi-beam dynamical electron diffraction theory with the column approximation to simulate the [0 0 1] zone-axis bright-field images of the QDs. Through comparison of experimental and simulated images, we showed composition segregation within the coherent Ge(Si)/Si(0 0 1) islands. In this paper, we investigate in detail how parameters of specimen thickness and electron accelerating voltage influence the [0 0 1] zone-axis bright-field images of Ge(Si)/Si(0 0 1) QDs, and we show the power of this technique for their detailed analysis.
2. Experiment and calculation Ge coherent islands were grown on Si(0 0 1) by solid source molecular beam epitaxy at a growth temperature of 700 C. Growth at this high temperature results in strong alloying effects of the substrate material (Si) into the Ge islands [20,21]. Details on the growth conditions have been described elsewhere [22]. Plan-view TEM specimens were prepared using chemical etching with a solution of HF and HNO3 in the ratio of 1:9. Plan-view TEM observations were carried out using a Philips EM430 operating at 300 kV. Following the procedures of our previous work [5], image simulations were carried out using multi-beam dynamical electron scattering theory with the column approximation, and absorption was included using the perturbation method and by assuming the imaginary parts of Fourier coefficients of lattice potential to be one-tenth of their real counterparts [23]. Calculations with different values of absorption parameters showed no significant differences to the images, except changes in relative contrast. During the image simulations, the sample is oriented with the QD downward in the microscope, to meet the experimental conditions and to make the programming easier. The displacement field was determined by finite element analysis using commercial software STRAND6 [24]. Pyramids with {5 0 1} facets and domes with steeper side facets have been the most frequently reported Ge(Si) island shapes [25]. The Ge(Si) islands observed in this investigation have been
ARTICLE IN PRESS X.Z. Liao et al. / Ultramicroscopy 98 (2004) 239–247
previously reported to be dome shaped with an aspect ratio of height-to-base diameter between 15 and 14 [19,20]. In the finite element analysis of the strain field, the island shape is simplified to be a spherical cap with an aspect ratio of 14: Fig. 1 shows the finite element model in cross-section. The fine lines in the model are the finite element meshes used in the calculation. In this model, the half width of the QD island (OA) has been set at 50 nm and equal to 4R=5 (R is the radius of the sphere) and the height (OB) has been set at 25 nm and equal to 2R=5; so that the height-to-base diameter ratio is 1:4 to match the experimentally determined values of island base diameter and aspect ratios. Previous comparisons between experimental and simulated images from these QDs have shown that there is a composition gradient within the QD, with a higher concentration of Ge at the top of the QD [19]. The calculations in this paper assume this same segregation model (except where stated otherwise). Three-dimensional models were generated by rotating the two-dimensional section in Fig. 1 through 90 around [0 0 1]. The geometry is meshed with eight-node elements in the substrate and with six-node triangular elements in the island. Because the model has axial symmetry around [0 0 1], and because the crystal lattice of the sample has fourfold symmetry, only one-quarter of the model needs to be calculated. Boundary
Fig. 1. The model of the (1 0 0) cross-section of a spherical-capshaped island. Only half of the model is shown as it is symmetrical about the axis [0 0 1] (OB). The dimension of CD is adjustable depending on the thickness of the substrate.
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conditions are set by the fourfold symmetry of the strain field, and by assuming that QDs are periodically arrayed in [1 0 0] and [0 1 0] directions. The lattice mismatch (f ) between the island and the substrate is represented by assuming the thermal expansion coefficients are f ðK1 Þ for the island and 0 ðK1 Þ for the substrate, and then by raising the temperature by 1 K. For the image simulations, column approximation calculations were performed for the square simulation area (with square edges parallel to /1 0 0S) of 2d 2d (where d is the base diameter of the model QDs). The number of the electron beams involved in the image simulations must be 4n þ 1; where n ¼ 1; 2; 3;y Test simulations show that the simulated images remain unchanged when the number of the electron beams is larger than 21 (and indeed there is very little change above n ¼ 2). In this paper, 21 electron beams were used.
3. Results Fig. 2 shows a plan-view [0 0 1] zone-axis brightfield diffraction contrast image of Ge(Si) coherent islands. The image background of white-and-black stripes is the thickness fringes, with the thinnest area in Fig. 2 located at the left bottom corner. The islands are relatively uniform in size with the
Fig. 2. An experimental plan-view [0 0 1] on-zone bright-field diffraction contrast image of coherent Ge(Si)/Si(0 0 1) islands. Four different kinds of contrast are marked as a, b, c and d. The arrows indicate /1 1 0S directions. Thickness fringes are seen on the background of the image and a scale bar is plotted showing the phase of the fringes.
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base diameter measured along /1 1 0S of about 100 nm. Four kinds of diffraction contrast are seen, as marked with a, b, c, and d. The same contrast appears periodically with the period of the thickness fringes. For example, contrast c appears in white fringes while contrast d appears at the middle of black fringes. This implies that the island diffraction contrast is a function of the specimen thickness and is related to the extinction distance of the electron beam employed. To understand how the sample thickness affects the diffraction contrast, the effective extinction distance of the (0 0 0) transmitted beam under [0 0 1] zone-axis diffraction conditions is shown in Fig. 3, for four different accelerating voltages. The
effective extinction distances are approximately 22 nm (100 kV), 30 nm (200 kV), 37.5 nm (300 kV) and 45 nm (400 kV). Bright field diffraction contrast images for 300 kV electrons were calculated for a range of TEM substrate thicknesses chosen to span a few periods of the extinction oscillations. Fig. 4 shows examples of the simulated images. It is seen that the images for the substrate thicknesses of 35 and 72.5 nm are very similar, and so are those for 47.5 and 85 nm, and also for 57.5 and 95 nm. That is, the images show a periodicity of 37.5 nm, in agreement with the extinction distance of Fig. 3c. In addition, images 4a and 4d, which, from Fig. 3 correspond to the maxima in the thickness fringes,
Fig. 3. The calculated intensity of the transmitted (0 0 0) electron beam vs. the silicon substrate thickness under the condition of [0 0 1] zone-axis diffraction and 21 electron beams. The electron accelerating voltage: (a) 100 kV, (b) 200 kV, (c) 300 kV and (d) 400 kV. Inserted in (c) is a scale bar corresponding to the one plotted in Fig. 2.
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Fig. 4. Simulated images for a coherent Ge(Si)/Si(0 0 1) island under 300 kV electrons and different Si substrate thickness. The substrate thicknesses are shown at the top of each image. The periodicity of the images as a function of electron extinction distance is clear. A /1 1 0S direction is vertical.
are very similar to the experimental images c in Fig. 2, which themselves are at the maxima of the thickness fringes in that image. Similarly images d in Fig. 2 are at minima of the thickness fringes, in agreement with their corresponding images 4c and 4f. We now consider the images in more detail. To do this, we plot in Fig. 2 the phase of the thickness fringes, with the maxima being at 0 (360 ) and the minima at 180 . Note that the thinnest part of the sample is at the bottom left corner. From this plot, it is seen that the image of type a occurs at 40–80 , image b at 80–120 , images c at 20 (340 )– +20 , and images d at 160–200 . Note that the lateral dimension of the coherent QDs is relatively large compared with the dimension of the thickness fringes and therefore the estimated phases of each type of images have a relatively large range. These values are entered in Table 1. The experimental images marked a, b, c, and d in Fig. 2 are enlarged in Fig. 5 ðaE Þ2ðdE Þ (the first row), respectively. While these four images are similar to one another, there are a number of significant differences: (1) the image aE is blurred and the coupled bars are broadest at the point marked by two white arrows, whereas in bE and dE they are broadest at the edges; (2) although both
Table 1 Comparison of the phases of the substrate thickness-fringes for experimental and simulated QD images. The phases in the last column are converted from the substrate thicknesses of the simulations Image label
Experimental estimated phase (degree)
Substrate thickness in simulation (nm)
Converted to phase (degree)
a b c
40–80 80–120 20 (340) B+20 160–200
44 47.5 72.5
66 99 336
54
161
d
bE and dE show a distinct double-cross with diverging coupled bars, the centres of the images are different: bE has a circular ring while dE has four black spots (one of the black spots is marked with a white arrow); (3) while most of the QD in cE has dark contrast, there are four brighter spots arranged with four fold symmetry at the centre of the image. Simulations were carried out with varying substrate thicknesses over steps of 2.5 nm, noting from the results above that the images repeat
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Fig. 5. Comparison of experimental and simulated images using 300 kV electrons. The first row shows enlarged experimental images of Fig. 2 (reoriented so that a /1 1 0S direction is vertical). Some special image features discussed in the text are high-lighted with white arrows. The second row shows simulated images with the substrate thicknesses shown at the top of each image.
periodically with thickness. Figs. 5 ðaS Þ2ðdS Þ (the second row), show the images which best fit aE to dE ; respectively. It can be seen that all of the image features (1) to (3) discussed above are found in the simulated images. The substrate thicknesses for these simulations can be converted to phase angles from Fig. 3, and their values are entered in Table 1. Remarkably good agreement is observed between the experimental and the simulated phase angles, showing how well the experimental and simulated images agree. This level of agreement cannot be found with QD models that have no segregation. It is clear that images in each column of Fig. 5 have the same image features. The excellent match of the experimental and simulated images shows that the image simulation method is a powerful tool for understanding the diffraction contrast of QDs. Different research groups have used different accelerating voltages to investigate QDs. For . et al. [26] and Sakamoto et al. example, Wohl [27] used 200 kV to investigate QDs in Ge/Si(0 0 1) and obtained similar double cross diffraction contrast to that reported here. Consequently, it is important to know how the electron accelerating voltage affects the diffraction contrast. To do this, images were simulated with voltages of 100, 200 and 400 kV. Figs. 6 and 7 show simulated images within a period of the extinction distances over thickness intervals of 5 nm for 200 and 400 kV, respectively.
Fig. 8 shows four simulated images for 100 kV electrons, spanning thicknesses over one period of the thickness fringe oscillation (detailed simulations with changing the substrate thickness show no double-cross contrast for 100 kV, unlike the case for 300 kV). From these results, it is seen that there is agreement between the periodicity of the simulated images and the periods shown by the thickness fringes in Fig. 3 for all the voltages studied. A common point for all voltages is that a QD appears as a dark island with little contrast at the phase angle of ðn 360Þ ; while the double cross contrast can be seen at the substrate thicknesses where the phase angle is close to [(n þ 0:5) 360] .
4. The effect of segregation on the images For the model QD with elemental segregation used in the above simulations, images with double crosses such as those shown in Figs. 4, 6 and 7 all have the bars of the double-cross approximately parallel to /1 1 0S and the coupled bars diverging from the image centre. The divergence of the coupled-bar is the key feature which leads to the conclusion of elemental segregation [19], since the bars do not diverge if the dot has no segregation. To test whether this same effect is observed for different voltages, simulations were carried out at 200 kV and 400 kV without segregation, and for
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Fig. 6. Simulated images for a coherent Ge(Si)/Si(0 0 1) island using 200 kV electrons and different Si substrate thicknesses. A /1 1 0S direction is vertical. The substrate thicknesses, with an interval of 5 nm, are shown at the top of each image.
Fig. 7. Simulated images for a coherent Ge(Si)/Si(0 0 1) island using 400 kV electrons and different Si substrate thicknesses. A /1 1 0S direction is vertical. The substrate thicknesses, with an interval of 5 nm, are shown at the top of each image.
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different concentrations of Ge which correspond to the island/substrate lattice mismatches of 2%, 3% and 4%. The results are shown in Fig. 9. It is evident by comparing Fig. 9 (200 and 400 kV
without segregation) with Figs. 6 and 7 (200 and 400 kV both with segregation) that, in all cases, no bar divergence is observed if segregation is absent. Moreover, as previously reported for 300 kV, for both voltages the images show a distinct doublecross with bars along /1 1 0S with the distance between the coupled bars decreasing as the lattice mismatch is increased. For 100 kV, the images for these QD sizes do not show double crosses, and so the study of segregation using the bar divergence cannot be made at this voltage.
5. Summary
Fig. 8. Simulated images for a coherent Ge(Si)/Si(0 0 1) island using 100 kV electrons and different Si substrate thicknesses. A /1 1 0S direction is vertical. The substrate thicknesses, with an interval of 5 nm, are shown at the top of each image.
The TEM [0 0 1] zone-axis bright-field diffraction contrast images of coherent Ge(Si)/Si(0 0 1) islands were investigated using multi-beam dynamical electron diffraction theory with the column approximation. Both experimental and simulated images show the diffraction contrast changing periodically with the Si TEM sample substrate thickness, and with the periodicity being the same as the effective extinction distances of (0 0 0). The image simulation is robust over a wide range of conditions, and the segregation effects within the QD can be studied by careful investigation of the image features.
Fig. 9. Simulated images for coherent Ge(Si)/Si(0 0 1) islands without composition segregation and with lattice mismatch of 4% [(a) and (d)], 3% [(b) and (e)], and 2% [(c) and (f)] and accelerating voltages of 200 kV [(a)–(c)] and 400 kV [(d)–(f)]. A /1 1 0S direction is vertical. The Si substrate thickness is 105 nm for both the 200 and 400 kV simulations.
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Acknowledgements The Ge(Si)/Si(0 0 1) sample was provided by Professor Z. M. Jiang of Fudan University, China. This work is supported by the Australian Research Council.
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[12] S.L. Sass, T. Mura, J.B. Cohen, Philos. Mag. 16 (1967) 679. [13] H.P. Degischer, Philos. Mag. 26 (1972) 1137. [14] S. Matsumura, M. Toyohara, Y. Tomokiyo, Philos. Mag. A 62 (1990) 653. [15] D. Lepski, Phys. Stat. Sol. (a) 24 (1974) 99. [16] I.N. Stranski, L. Krastanow, Sitzungsber. Akad. Wiss. Wien Abt. IIb Band Wiss. Lit. 146 (1937) 797. [17] S. Christiansen, M. Albrecht, H.P. Strunk, H.J. Maier, Appl. Phys. Lett. 64 (1994) 3617. [18] K. Scheerschmidt, D. Conrad, H. Kirmse, R. Schneider, W. Neumann, Ultramicroscopy 81 (2000) 289. [19] X.Z. Liao, J. Zou, D.J.H. Cockayne, Z.M. Jiang, X. Wang, J. Appl. Phys. 90 (2001) 2725. [20] X.Z. Liao, J. Zou, D.J.H. Cockayne, J. Qin, Z.M. Jiang, X. Wang, R. Leon, Phys. Rev. B 60 (1999) 15605. [21] X.Z. Liao, J. Zou, D.J.H. Cockayne, Z.M. Jiang, X. Wang, R. Leon, App. Phys. Lett. 77 (2000) 1304. [22] X. Wang, Z.M. Jiang, H.J. Zhu, F. Lu, D.M. Huang, X.H. Liu, C.W. Hu, Y.F. Chen, Z.Q. Zhu, T. Yao, Appl. Phys. Lett. 71 (1997) 3543. [23] C.J. Humphreys, Rep. Prog. Phys. 42 (1979) 1825. [24] http://www.strand.aust.com [25] G. Medeiros-Ribeiro, A.M. Bratkovski, T.I. Kamins, D.A.A. Ohlberg, R.S. Williams, Science 279 (1998) 353. . . [26] G. Wohl, C. Schollhorn, O.G. Schmidt, K. Brunner, K. Eberl, O. Kienzle, F. Ernst, Thin Solid Films 321 (1998) 86. [27] K. Sakamoto, H. Matsuhata, M.O. Tanner, D. Wang, K.L. Wang, Thin Solid Films 321 (1988) 55.
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[0 0 1] zone-axis bright-field diffraction contrast from coherent Ge(Si) islands on Si(0 0 1) X.Z. Liaoa,b, J. Zoua,c,*, D.J.H. Cockayned, S. Matsumurae a
Australian Key Centre for Microscopy and Microanalysis, The University of Sydney, Sydney, NSW 2006, Australia b Division of Materials Science and Technology, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Division of Materials and Centre for Microscopy and Microanalysis, The University of Queensland, QLD 4072, Australia d Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK e Department of Applied Quantum Physics and Nuclear Engineering, Kyushu University, Fukuoka 812-8581, Japan Received 14 April 2003; received in revised form 8 July 2003 Dedicated to Professor Fang-hua Li on the occasion of her 70th birthday
Abstract Coherent Ge(Si)/Si(0 0 1) quantum dot islands grown by solid source molecular beam epitaxy at a growth temperature of 700 C were investigated using transmission electron microscopy working at 300 kV. The [0 0 1] zone-axis bright-field diffraction contrast images of the islands show strong periodicity with the change of the TEM sample substrate thickness and the period is equal to the effective extinction distance of the transmitted beam. Simulated images based on finite element models of the displacement field and using multi-beam dynamical diffraction theory show a high degree of agreement. Studies for a range of electron energies show the power of the technique for investigating composition segregation in quantum dot islands. r 2003 Elsevier B.V. All rights reserved. PACS: 68.37.Lp; 81.07.Ta; 81.05.Cy Keywords: TEM characterization; Image simulation; Quantum dot
1. Introduction Coherent hetero-epitaxial grown semiconductor islands such as Ge(Si)/Si [1] and InGaAs/GaAs [2] are of great interest because of their potential applications as semiconductor quantum dots *Corresponding author. Centre for Microscopy and Microanalysis, The University of Queensland, Queensland, QLD 4072, Australia. Tel.: +61-7-336-53977; fax: +61-7-336-54422. E-mail address: [email protected] (J. Zou).
(QDs). The opto-electronic properties of QDs depend strongly on the structural parameters including the shape, size, composition and strain distribution of the QDs [3,4]. Considerable effort has been made to investigate these structural parameters, with transmission electron microscopy (TEM) diffraction contrast being frequently used. For example, Liao et al. [5] investigated the composition of InGaAs/GaAs(0 0 1) QDs using the [0 0 1] multi-beam bright-field diffraction contrast imaging technique; Liu et al. [6] measured the
0304-3991/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2003.08.017
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strain evolution of coherent Ge islands on Si(0 0 1) using a TEM technique based on two-beam darkfield diffraction contrast imaging; and Benabbas et al. [7] and Androussi et al. [8] studied the shape and size of InAs/GaAs QDs by applying two-beam dynamical electron diffraction contrast imaging techniques. TEM diffraction contrast has been used for the investigation of coherent precipitates since the early 1960s when Ashby and Brown [9,10] showed that diffraction contrast from inclusions could be used to yield information about the magnitude of the strain. Image simulation is an important tool for comparing different strain models with experimental images and obtaining a reliable strain field distribution from a model structure is the important first step. Some early simulations of displacement fields of coherent precipitates were carried out within the framework of isotropic elasticity theory [9–11] while later simulations used anisotropic elasticity [12–15]. These early investigations calculated the strain fields using analytical expressions, but this is only possible for a few simple regular shapes such as spheres [13] and cubes [12]. However, with the increasing interest in Stranski-Krastanow [16] grown semiconductor QDs (which can be regarded as coherent precipitates), obtaining strain field distributions for more complicated shapes such as lenses or pyramids, and for unburied ‘‘precipitates’’, has become increasingly important. The improving speed of numerical computations has made it possible to calculate the displacement field for strained systems such as QDs using numerical methods rather than analytical expressions. For example, Christiansen et al. [17] used finite element analysis to obtain the displacement field of Ge(Si) islands grown on Si. Their study showed that the displacement field obtained in this way correctly explains features of convergent beam diffraction patterns. Scheerschmidt et al. [18] used molecular dynamics to derive the displacement field for CdSe/ZnSe QDs, and applied the result to the simulation of TEM images. In our earlier investigation, we have successfully employed finite element analysis to generate the strain field of unburied Ge(Si)/Si(0 0 1) [19] QD
islands and used the multi-beam dynamical electron diffraction theory with the column approximation to simulate the [0 0 1] zone-axis bright-field images of the QDs. Through comparison of experimental and simulated images, we showed composition segregation within the coherent Ge(Si)/Si(0 0 1) islands. In this paper, we investigate in detail how parameters of specimen thickness and electron accelerating voltage influence the [0 0 1] zone-axis bright-field images of Ge(Si)/Si(0 0 1) QDs, and we show the power of this technique for their detailed analysis.
2. Experiment and calculation Ge coherent islands were grown on Si(0 0 1) by solid source molecular beam epitaxy at a growth temperature of 700 C. Growth at this high temperature results in strong alloying effects of the substrate material (Si) into the Ge islands [20,21]. Details on the growth conditions have been described elsewhere [22]. Plan-view TEM specimens were prepared using chemical etching with a solution of HF and HNO3 in the ratio of 1:9. Plan-view TEM observations were carried out using a Philips EM430 operating at 300 kV. Following the procedures of our previous work [5], image simulations were carried out using multi-beam dynamical electron scattering theory with the column approximation, and absorption was included using the perturbation method and by assuming the imaginary parts of Fourier coefficients of lattice potential to be one-tenth of their real counterparts [23]. Calculations with different values of absorption parameters showed no significant differences to the images, except changes in relative contrast. During the image simulations, the sample is oriented with the QD downward in the microscope, to meet the experimental conditions and to make the programming easier. The displacement field was determined by finite element analysis using commercial software STRAND6 [24]. Pyramids with {5 0 1} facets and domes with steeper side facets have been the most frequently reported Ge(Si) island shapes [25]. The Ge(Si) islands observed in this investigation have been
ARTICLE IN PRESS X.Z. Liao et al. / Ultramicroscopy 98 (2004) 239–247
previously reported to be dome shaped with an aspect ratio of height-to-base diameter between 15 and 14 [19,20]. In the finite element analysis of the strain field, the island shape is simplified to be a spherical cap with an aspect ratio of 14: Fig. 1 shows the finite element model in cross-section. The fine lines in the model are the finite element meshes used in the calculation. In this model, the half width of the QD island (OA) has been set at 50 nm and equal to 4R=5 (R is the radius of the sphere) and the height (OB) has been set at 25 nm and equal to 2R=5; so that the height-to-base diameter ratio is 1:4 to match the experimentally determined values of island base diameter and aspect ratios. Previous comparisons between experimental and simulated images from these QDs have shown that there is a composition gradient within the QD, with a higher concentration of Ge at the top of the QD [19]. The calculations in this paper assume this same segregation model (except where stated otherwise). Three-dimensional models were generated by rotating the two-dimensional section in Fig. 1 through 90 around [0 0 1]. The geometry is meshed with eight-node elements in the substrate and with six-node triangular elements in the island. Because the model has axial symmetry around [0 0 1], and because the crystal lattice of the sample has fourfold symmetry, only one-quarter of the model needs to be calculated. Boundary
Fig. 1. The model of the (1 0 0) cross-section of a spherical-capshaped island. Only half of the model is shown as it is symmetrical about the axis [0 0 1] (OB). The dimension of CD is adjustable depending on the thickness of the substrate.
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conditions are set by the fourfold symmetry of the strain field, and by assuming that QDs are periodically arrayed in [1 0 0] and [0 1 0] directions. The lattice mismatch (f ) between the island and the substrate is represented by assuming the thermal expansion coefficients are f ðK1 Þ for the island and 0 ðK1 Þ for the substrate, and then by raising the temperature by 1 K. For the image simulations, column approximation calculations were performed for the square simulation area (with square edges parallel to /1 0 0S) of 2d 2d (where d is the base diameter of the model QDs). The number of the electron beams involved in the image simulations must be 4n þ 1; where n ¼ 1; 2; 3;y Test simulations show that the simulated images remain unchanged when the number of the electron beams is larger than 21 (and indeed there is very little change above n ¼ 2). In this paper, 21 electron beams were used.
3. Results Fig. 2 shows a plan-view [0 0 1] zone-axis brightfield diffraction contrast image of Ge(Si) coherent islands. The image background of white-and-black stripes is the thickness fringes, with the thinnest area in Fig. 2 located at the left bottom corner. The islands are relatively uniform in size with the
Fig. 2. An experimental plan-view [0 0 1] on-zone bright-field diffraction contrast image of coherent Ge(Si)/Si(0 0 1) islands. Four different kinds of contrast are marked as a, b, c and d. The arrows indicate /1 1 0S directions. Thickness fringes are seen on the background of the image and a scale bar is plotted showing the phase of the fringes.
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base diameter measured along /1 1 0S of about 100 nm. Four kinds of diffraction contrast are seen, as marked with a, b, c, and d. The same contrast appears periodically with the period of the thickness fringes. For example, contrast c appears in white fringes while contrast d appears at the middle of black fringes. This implies that the island diffraction contrast is a function of the specimen thickness and is related to the extinction distance of the electron beam employed. To understand how the sample thickness affects the diffraction contrast, the effective extinction distance of the (0 0 0) transmitted beam under [0 0 1] zone-axis diffraction conditions is shown in Fig. 3, for four different accelerating voltages. The
effective extinction distances are approximately 22 nm (100 kV), 30 nm (200 kV), 37.5 nm (300 kV) and 45 nm (400 kV). Bright field diffraction contrast images for 300 kV electrons were calculated for a range of TEM substrate thicknesses chosen to span a few periods of the extinction oscillations. Fig. 4 shows examples of the simulated images. It is seen that the images for the substrate thicknesses of 35 and 72.5 nm are very similar, and so are those for 47.5 and 85 nm, and also for 57.5 and 95 nm. That is, the images show a periodicity of 37.5 nm, in agreement with the extinction distance of Fig. 3c. In addition, images 4a and 4d, which, from Fig. 3 correspond to the maxima in the thickness fringes,
Fig. 3. The calculated intensity of the transmitted (0 0 0) electron beam vs. the silicon substrate thickness under the condition of [0 0 1] zone-axis diffraction and 21 electron beams. The electron accelerating voltage: (a) 100 kV, (b) 200 kV, (c) 300 kV and (d) 400 kV. Inserted in (c) is a scale bar corresponding to the one plotted in Fig. 2.
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Fig. 4. Simulated images for a coherent Ge(Si)/Si(0 0 1) island under 300 kV electrons and different Si substrate thickness. The substrate thicknesses are shown at the top of each image. The periodicity of the images as a function of electron extinction distance is clear. A /1 1 0S direction is vertical.
are very similar to the experimental images c in Fig. 2, which themselves are at the maxima of the thickness fringes in that image. Similarly images d in Fig. 2 are at minima of the thickness fringes, in agreement with their corresponding images 4c and 4f. We now consider the images in more detail. To do this, we plot in Fig. 2 the phase of the thickness fringes, with the maxima being at 0 (360 ) and the minima at 180 . Note that the thinnest part of the sample is at the bottom left corner. From this plot, it is seen that the image of type a occurs at 40–80 , image b at 80–120 , images c at 20 (340 )– +20 , and images d at 160–200 . Note that the lateral dimension of the coherent QDs is relatively large compared with the dimension of the thickness fringes and therefore the estimated phases of each type of images have a relatively large range. These values are entered in Table 1. The experimental images marked a, b, c, and d in Fig. 2 are enlarged in Fig. 5 ðaE Þ2ðdE Þ (the first row), respectively. While these four images are similar to one another, there are a number of significant differences: (1) the image aE is blurred and the coupled bars are broadest at the point marked by two white arrows, whereas in bE and dE they are broadest at the edges; (2) although both
Table 1 Comparison of the phases of the substrate thickness-fringes for experimental and simulated QD images. The phases in the last column are converted from the substrate thicknesses of the simulations Image label
Experimental estimated phase (degree)
Substrate thickness in simulation (nm)
Converted to phase (degree)
a b c
40–80 80–120 20 (340) B+20 160–200
44 47.5 72.5
66 99 336
54
161
d
bE and dE show a distinct double-cross with diverging coupled bars, the centres of the images are different: bE has a circular ring while dE has four black spots (one of the black spots is marked with a white arrow); (3) while most of the QD in cE has dark contrast, there are four brighter spots arranged with four fold symmetry at the centre of the image. Simulations were carried out with varying substrate thicknesses over steps of 2.5 nm, noting from the results above that the images repeat
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Fig. 5. Comparison of experimental and simulated images using 300 kV electrons. The first row shows enlarged experimental images of Fig. 2 (reoriented so that a /1 1 0S direction is vertical). Some special image features discussed in the text are high-lighted with white arrows. The second row shows simulated images with the substrate thicknesses shown at the top of each image.
periodically with thickness. Figs. 5 ðaS Þ2ðdS Þ (the second row), show the images which best fit aE to dE ; respectively. It can be seen that all of the image features (1) to (3) discussed above are found in the simulated images. The substrate thicknesses for these simulations can be converted to phase angles from Fig. 3, and their values are entered in Table 1. Remarkably good agreement is observed between the experimental and the simulated phase angles, showing how well the experimental and simulated images agree. This level of agreement cannot be found with QD models that have no segregation. It is clear that images in each column of Fig. 5 have the same image features. The excellent match of the experimental and simulated images shows that the image simulation method is a powerful tool for understanding the diffraction contrast of QDs. Different research groups have used different accelerating voltages to investigate QDs. For . et al. [26] and Sakamoto et al. example, Wohl [27] used 200 kV to investigate QDs in Ge/Si(0 0 1) and obtained similar double cross diffraction contrast to that reported here. Consequently, it is important to know how the electron accelerating voltage affects the diffraction contrast. To do this, images were simulated with voltages of 100, 200 and 400 kV. Figs. 6 and 7 show simulated images within a period of the extinction distances over thickness intervals of 5 nm for 200 and 400 kV, respectively.
Fig. 8 shows four simulated images for 100 kV electrons, spanning thicknesses over one period of the thickness fringe oscillation (detailed simulations with changing the substrate thickness show no double-cross contrast for 100 kV, unlike the case for 300 kV). From these results, it is seen that there is agreement between the periodicity of the simulated images and the periods shown by the thickness fringes in Fig. 3 for all the voltages studied. A common point for all voltages is that a QD appears as a dark island with little contrast at the phase angle of ðn 360Þ ; while the double cross contrast can be seen at the substrate thicknesses where the phase angle is close to [(n þ 0:5) 360] .
4. The effect of segregation on the images For the model QD with elemental segregation used in the above simulations, images with double crosses such as those shown in Figs. 4, 6 and 7 all have the bars of the double-cross approximately parallel to /1 1 0S and the coupled bars diverging from the image centre. The divergence of the coupled-bar is the key feature which leads to the conclusion of elemental segregation [19], since the bars do not diverge if the dot has no segregation. To test whether this same effect is observed for different voltages, simulations were carried out at 200 kV and 400 kV without segregation, and for
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Fig. 6. Simulated images for a coherent Ge(Si)/Si(0 0 1) island using 200 kV electrons and different Si substrate thicknesses. A /1 1 0S direction is vertical. The substrate thicknesses, with an interval of 5 nm, are shown at the top of each image.
Fig. 7. Simulated images for a coherent Ge(Si)/Si(0 0 1) island using 400 kV electrons and different Si substrate thicknesses. A /1 1 0S direction is vertical. The substrate thicknesses, with an interval of 5 nm, are shown at the top of each image.
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different concentrations of Ge which correspond to the island/substrate lattice mismatches of 2%, 3% and 4%. The results are shown in Fig. 9. It is evident by comparing Fig. 9 (200 and 400 kV
without segregation) with Figs. 6 and 7 (200 and 400 kV both with segregation) that, in all cases, no bar divergence is observed if segregation is absent. Moreover, as previously reported for 300 kV, for both voltages the images show a distinct doublecross with bars along /1 1 0S with the distance between the coupled bars decreasing as the lattice mismatch is increased. For 100 kV, the images for these QD sizes do not show double crosses, and so the study of segregation using the bar divergence cannot be made at this voltage.
5. Summary
Fig. 8. Simulated images for a coherent Ge(Si)/Si(0 0 1) island using 100 kV electrons and different Si substrate thicknesses. A /1 1 0S direction is vertical. The substrate thicknesses, with an interval of 5 nm, are shown at the top of each image.
The TEM [0 0 1] zone-axis bright-field diffraction contrast images of coherent Ge(Si)/Si(0 0 1) islands were investigated using multi-beam dynamical electron diffraction theory with the column approximation. Both experimental and simulated images show the diffraction contrast changing periodically with the Si TEM sample substrate thickness, and with the periodicity being the same as the effective extinction distances of (0 0 0). The image simulation is robust over a wide range of conditions, and the segregation effects within the QD can be studied by careful investigation of the image features.
Fig. 9. Simulated images for coherent Ge(Si)/Si(0 0 1) islands without composition segregation and with lattice mismatch of 4% [(a) and (d)], 3% [(b) and (e)], and 2% [(c) and (f)] and accelerating voltages of 200 kV [(a)–(c)] and 400 kV [(d)–(f)]. A /1 1 0S direction is vertical. The Si substrate thickness is 105 nm for both the 200 and 400 kV simulations.
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Acknowledgements The Ge(Si)/Si(0 0 1) sample was provided by Professor Z. M. Jiang of Fudan University, China. This work is supported by the Australian Research Council.
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