The composition dependent mechanical properties of ge si core–shell nanowires

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Physica E 40 (2008) 3042–3048 The composition-dependent mechanical properties of Ge/Si core–shell nanowires X.W. Liu, J. Hu, B.C. Pan à Hefei National Laboratory for Physical Sciences at Microscale, Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, PR China Received 7 November 2007; received in revised form 14 March 2008; accepted 25 March 2008 Available online 12 April 2008 Abstract The Stillinger–Weber potential is used to study the composition-dependent Young’s modulus for Ge-core/Si-shell and Si-core/Ge-shell nanowires. Here, the composition is defined as a ratio of the number of atoms of the core to the number of atoms of a core–shell nanowire. For each concerned Ge-core/Si-shell nanowire with a specified diameter, we find that its Young’s modulus increases to a maximal value and then decreases as the composition increases. Whereas Young’s modulus of the Si-core/Ge-shell nanowires increase nonlinearly in a wide compositional range. Our calculations reveal that these observed trends of Young’s modulus of core–shell nanowires are essentially attributed to the different components of the cores and the shells, as well as the different strains in the interfaces between the cores and the shells. r 2008 Elsevier B.V. All rights reserved. PACS: 61.46.Àw; 11.15.Kc; 46.80.þj; 74.62.Dh Keywords: Nanowires; Mechanical properties; Calculation 1. Introduction There has been fast-growing interest in semiconductor nanowires due to their unique size- dependent electronic, optical and transport properties [1–4]. Among various kinds of nanowires, composite nanowires, where their sizes and the composition can be modulated, provide potential applications in thermoelectronics, nanoelectronics and optoelectronics [5–7]. Therefore, much effort has focused on the synthesis of core–shell nanowires with different compositions in the past few years [8–10]. Typically, core– shell nanowires consisting of germanium and silicon have been synthesized using chemical vapor deposition method successfully [5]. Later on, Musin and Wang [11, 12] studied the composition- and size-dependent band gaps of Ge- core/Si-shell and Si-core/Ge-shell nanowires at the level of density functional theory, where the composition is defined as the ratio of the number of atoms of the core to the number of the atoms of the whole nanowire. They found that the band gaps of both kinds of core–shell nanowires decrease when composition o0:3 and increase after that. However for a given composition, the band gap decreases noticeably as the diameter of the nanowire increases. Therefore, the Ge-core/Si-shell and Si-core/Ge-shell nano- wires exhibit promising perspective for band gap engineer- ing and optoelectronic applications [11–13]. Basically, for most of novel materials, their mechanical properties are of great importance for their potential application. Since so, assessing the mechanical properties of such core–shell nanowires is necessary. Usually, the mechanical property of a nanowire can be described by Young’s modulus. Previous publications showed that Young’s modulus of nanoscale structure relies upon the size of the structure and the orientation of lateral facets. For example, the Young’s modulus of single-crystal GaN nanotube increases as the ratio of the surface area to its volume grows; Young’s modulus of single-crystal GaN nanotube oriented along ½110 is higher than that of ½001 oriented single-crystal GaN nanotube [14]. The similar ARTICLE IN PRESS www.elsevier.com/locate/physe 1386-9477/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.03.011 à Corresponding author. E-mail address: bcpan@ustc.edu.cn (B.C. Pan). results were also observed for the case of ZnO nanowires in experiment [15]. Although there have been an increasing work devoted to Young’s modulus of nanowires [16–18], there are few studies on Young’s modulus of core–shell nanowires up to now. In this work, we perform our calculations on Ge-core/Si- shell and Si-core/Ge-shell nanowires to evaluate their Young’s modulus. we find that Young’s modulus are dependent on the composition, which is explained using the strains at the interface between the cores and the shells. 2. Computational details Experiments showed that the cross sections of the produced Ge-core/Si-shell or Si-core/Ge-shell nanowires are all hexagonal [5]. Because of this, we initially generate core–shell nanowires with hexagonal cross section on the basis of diamond-structured crystals [19]. For example, a Si-core (that is a nanowire) orientated along ½111 direction is isolated from Si crystal. For convenience, the distance between the axis and the vertex of hexagon on the cross section is defined as the radius R core of a core as shown in Fig. 1. For a Ge-shell with inner radius R in shell and outer radius R out shell , it is generated using the scheme proposed in previous work [14]. By filling a Ge-shell with a suitable Si-core, we ach ieve a Si-core/Ge-shell nanowire. With using this scheme, Ge-core/Si-shell nanowires are also generated. Clearly, through adjusting the diameter of the core and the thickness of the shell, we can obtain various Si-core/Ge-shell and Ge-co re/Si-shell nanowires. For a given core–shell nanowire, the composition, N core =ðN core þ N shell Þ, where the N core and N shell are the number of core atoms and shell atoms, respectively, does actually reflect the structural feature described by ð R core /R core2shell Þ 2 .In order to reveal how the composition of a core–shell nanowire affects its Young’s modulus, we select a core– shell nanowire with a specified radius R core2shell of about 31 a ˚ , where the radius of the core (R core ) and the thickness of the shell ðT shell ¼ R out shell À R in shell Þ are adjustable with a limitation of ðR core þ T shell ¼ R core2shell Þ. Tables 1 and 2 list the structural features of the core–shell nanowires we considered. As listed in Tables 1 and 2, 20 core–shell nanowires are taken into account, each of which contains 2524 atoms. To fully optimize these large systems and study their mechan- ical properties, the proposed Stillinger–Weber (SW) potentials for Si, Ge and Ge–Si [20] are employed for our calculations, where the total energy of a system contains one-body, two-body and three-body contributions. The potential functions for the two-body and three-body are parameterized. By fitting to some bulk properties, the parameters for Si [20] and Ge [21] were, respectively, achieved. According to these parameters, the parameters for the Ge–Si were taken to be the geometric means of the both sets of parameters [22]. Previously, this empirical potential has been used to handle silicon–germanium ARTICLE IN PRESS Fig. 1. (Color online) Top view of a core–shell nanowire. The larger circles stand for the core atoms, and the smaller ones for the shell-atoms. Table 2 The optimal structural parameters of Si-core/Ge-shell nanowires Radius of core Inner radius of shell Outer radius of shell Number of atoms in core Composition 6.68 9.02 32.22 148 0.0586 8.91 11.25 32.18 244 0.0967 11.15 13.48 32.12 364 0.1442 13.37 15.71 32.05 508 0.2013 15.60 17.93 31.97 676 0.2678 17.83 20.15 31.87 868 0.3439 20.05 22.36 31.75 1084 0.4295 22.27 24.57 31.64 1324 0.5246 24.47 26.76 31.48 1588 0.6292 26.65 28.92 31.32 1876 0.7433 The number of atoms of core–shell nanowires are fixed to be 2524. The unit of the length is in a ˚ . Table 1 The optimal structural parameters of Ge-core/Si-shell nanowires Radius of core Inner radius of shell Outer radius of shell Number of atoms in core Composition 6.87 9.01 31.04 148 0.0586 9.16 11.29 31.08 244 0.0967 11.45 13.58 31.14 364 0.1442 13.74 15.87 31.20 508 0.2013 16.04 18.16 31.27 676 0.2678 18.33 20.46 31.36 868 0.3439 20.63 22.76 31.47 1084 0.4295 22.93 25.07 31.60 1324 0.5246 25.25 27.41 31.76 1588 0.6292 27.58 29.77 31.95 1876 0.7433 The number of atoms of core–shell nanowires are fixed to be 2524. The unit of the length is in a ˚ . X.W. Liu et al. / Physica E 40 (2008) 3042–3048 3043 alloys [22]. More recen tly, the Young’s modulus of Si nanowires have been studied based on this classical potential, which is in good agreement with the density functional theory calculations [16,17]. Such agreement shows that it is reliable to handle the Si/Ge core–shell nanowires using this potential. Commonly, Young’s modulus of a nanowire can be calculated according to the following expression: Y ¼ 1 V 0 q 2 E qe 2      e¼0 , (1) where E is the total energy, V 0 is the equilibrium volume, which is defined as the product of axial equilibrium lengt h ð‘ 0 Þ and the cross-section area S 0 . e is longitudinal strain. In our calculations, the periodic condition along the axis of each wire is imposed. Initially, the lattice constant corresponding to the ideal bulk Ge is employed for a concerned Si–Ge core–shell nanowire. Clearly, this lattice constant is not optimal. Then, we adjust the lattice constant of the nanowire. For each specified lattice constant, the nanowire is fully relaxed. We thus obtain the energies of the nanowire with different lattice constant. From these energies, the optimal lattice constant of the nanowire is achieved. Furthermore, the nanowire is elongated and compressed axially from À1.8% to 1.8% by increment of 0.3% around its equilibrium, to obtain an energy curve (the total energy of a system vs the loaded strain). This curve is fitted by using a cubic polynomial function [14]. Inserting the cubic polynomial function into Eq. (1), we obtain Young’s modulus of the nanowire. 3. Results and discussion Fig. 2(a) plots the calculated Young’s modulus of the core–shell nanowires with different compositions. It is found that in the case of Si-core/Ge-shell, the Young’s modulus increases with increasing the component of Si-core, whereas in the case of Ge-core/Si-shell, the Young’s modulus goes up and then decreases. Such an increasing-and-decre asing trend in Young’s modulus curve of Ge-core/Si-shell and monotonous increment in Young’s modulus of Si-core/Ge-shell strongly indicate that the mechanical property of a core–shell nanowire is dependent on its composition. Structurally, for a given core–shell nanowire, as the diameter of the core increases, the thickness of the shell decreases correspondingly. In a sense, the core and the shell may be analogous to the isolated wire and the isolated tube, respectively. Our calculations show that Young’s modulus of an isolated Si (Ge) nanowire increases as its diameter decreases, and Young’s modulus of an isolated Si (Ge) nanotube becomes large when its thickness is small (Fig. 3). It seems that the Young’s modulus of the core and the shell are competing each other to result in the composition-dependent trends of the Young’s modulus shown in Fig. 2(a). However, the core and the shell in a considered core–shell nanowire do couple with each other, and thus the Young’s modulus of either core or shell is not the same as that of the isolated nanowire or the isolated nanotube. On the other hand, the Young’s modulus of a core–shell nanowire can not be expressed as a simple summation of the Young’s modulus of the isolated nanowires and the isolated nanotubes. To explicitly reveal the relation of the Young’s modulus between a core–shell nanowire and its core and shell, we employ the ‘‘stress–strain relation’’ to serve our analysis. As we know, the stress s zz is proportional to the loaded strain e z along z direction for string-like materials [23], s zz ¼ Y e z . (2) The proportional coefficient Y above is the Young’s modulus along z direction. Here, the stress of the system ARTICLE IN PRESS 0.0 Composition 144.5 145.5 146.5 147.5 148.5 Young’s modulus (GPa) Ge−core/Si−shell Si−core/Ge−shell 0.0 Composition 144.5 145.5 146.5 147.5 148.5 Young’s modulus (GPa) Ge−core/Si−shell Si−core/Ge−shell 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 Fig. 2. (Color online) Young’s modulus of Ge-core/Si-shell and Si-core/ Ge-shell nanowires as a function of composition, obtained by using (a) formula (1), (b) the ‘‘stress–strain relation’’. X.W. Liu et al. / Physica E 40 (2008) 3042–30483044 is evaluated by [18] s zz ¼ 1 2V X i X N jai f z ij r z ij , (3) where V is the volume of the system, N is the number of atoms of a core–shell nanowire, f z ij is the inter-particle force along z direction between particles i and j, and r z ij is the spacing in z direction between the two particles. These variables are a function of strain. To calculate the stresses of the core and the shell under a loaded strain, t he right-hand s ide o f for mula (3) i s r ewritten as 1 2V X N core i¼1 X N jðaiÞ¼1 f z ij r z ij þ 1 2V X N i¼N core þ1 X N jðaiÞ¼1 f z ij r z ij The first term above contains the interaction between core-atom and core-atom and the interaction between core- atom and shell-atom, and the second term contains the interaction between shell-atom and shell-atom and between shell-atom and core-atom. The total stress of a core–shell nanowire can be rewritten as s zz ¼ 1 2V X core f z ij r z ij þ 1 2V X shell f z ij r z ij . (4) With defining the volumes of the core and the shell, V core and V shell , we yield s zz ¼ V core V s zz core þ V shell V s zz shell , (5) where s zz core and s zz shell are the stresses of the core and the shell, respectively. Considering expressions (2) and (5), we have Y core2shell ¼ V core V Y core þ V shell V Y shell . (6) From this formula, total Young’s modulus Y core2 shell is not only contributed from Young’s modulus of the core ðY core Þ and the shell (Y shell ) but also dependent on the fractional volumes of V core =V and V shell =V. That is, Young’s modulus of a whole core–shell nanowire is the weighted combination of Young’s modulus of the core and the shell. To evaluate Young’s modulus of the core and the shell using the ‘‘stress–strain relation’’, it is necessary to calculate their volumes. It is worth noting that the existence of the interface region between the core and the shell in a core–shell nanowire results in difficulty for defining volumes of the core and the shell. When the volume is taken to be the geometric volume, the calculated Young’s modulus of the core (shell) decreases with increasing its diameter (thickness) as plotted in Fig. 4, exhibiting the ARTICLE IN PRESS 0 Radius (Ang) Thickness of Nanotube (Ang) 144 146 148 150 152 Young’s modulus (GPa) Ge nanowire Si nanowire 10 20 30 40 158 156 154 152 150 148 146 144 4 81216 20 24 Young’s modulus (GPa) Ge nanotube Si nanotube Fig. 3. (Color online) Young’s modulus of silicon and germanium (a) nanowirs and (b) nanotubes, obtained by using formula (1). 0.0 Composition Young’s modulus (GPa) 140 160 180 200 220 240 Ge−core Si−shell Si−core Ge−shell 0.2 0.4 0.6 0.8 Fig. 4. (Color online) Young’s modulus of the cores and the shells obtained by using the ‘‘stress–strain relation’’. The geometric volumes for the cores and the shells are taken into account in calculations. X.W. Liu et al. / Physica E 40 (2008) 3042–3048 3045 same trend as that of the isolated nanowire (nanotube) as addressed above. The main discrepancy between Figs. 3 and 4 is the systematical shift-up of Young’s modulus evaluated by the ‘‘stress–strain relation’’ relative to that calculated by formula (1), this is essentially resulted from the interaction between the core and the shell in the core– shell nanowire. In fact, the volumes of the core and the shell should be ‘‘physical volumes’’, and thus the geometric volumes used above are not suitable for the case of the core–shell nanowire. Unfortunately, the definition for ‘‘physical volumes’’ of the core and the shell in a core– shell nanowire is somewhat uncertain, this is due to the existence of the region around the interface between the core and the shell of a core–shell nanowire. Basically, it is unreasonable for the core or the shell to include the volume of the whole interface region. A possible way is to divide the whole interface region into two parts according to the ratio of bondlengths of bulk Si and bulk Ge, and the two parts, respectively, belong to the core and the shell. In this case, Young’s modulus of Ge- core decreases as its diameter increases strikingly, while that of the Si-core goes up slowly (Fig. 5); The changes of Young’s modulus of shells with increasing composition are just opposite to the case of Ge-cores. This clearly indicates that the embedded Si nanowire in a Ge nanotube exhibits a unusual behavior in its Young’s modulus with respect to the isolated Si nanowire, while the embedded Ge nanowire in a Si nanotube just follows the normal trend in Young’s modulus. We emphasize that the behaviors above are essentially originated from the interface effect between the core and the shell, in which the volumes of the core and shell and the caused stresses around the interface play important roles. Firstly, let us pay our attention to the volume effect. Fig. 6(a) displays the fractional volumes of cores and shells as a function of the composition, from which we can find that as the composition increases, the fractional volumes of the cores linearly increase, while the fractional volumes of the shells linearly decrease. According to these fractional volumes and the obtained Young’s modulus (Y core and Y shell ), we immediately obtain the components, ðV core =VÞ Y core and ðV shell =VÞY shell , of Young’s modulus of the core– shell nanowires. Surprisingly, the two components of Young’s modulus as a function of the composition almost exhibit a linear trend (Fig. 6(b)), which is totally different from the trends of Y core and Y shell , but strikingly similar to the trends of the fractional volumes displayed in Fig. 6(a). This observation strongly demonstrates that the ‘‘physical ARTICLE IN PRESS 0.0 Composition 144 146 148 150 152 154 Young’s modulus (GPa) Ge−core Si−shell Si−core Ge−shell 0.2 0.4 0.6 0.8 Fig. 5. (Color online) Young’s modulus of the cores and the shells calculated by using the ‘‘stress–strain relation’’. The ‘‘physical volumes’’ as addressed in the text for the cores and the shells are taken into account in calculations. 0.0 Composition 0.0 0.2 0.4 0.6 0.8 1.0 Fractional volume Ge−core Si−shell si−core Ge−shell 0.0 Composition 0 50 100 150 Young’s modulus (GPa) Ge−core Si−shell Si−core Ge−shell 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Fig. 6. (Color online) (a) The fractional volumes of cores and shells vs the composition. (b) The components of the total Young’s modulus as a function of composition for each core–shell nanowire. X.W. Liu et al. / Physica E 40 (2008) 3042–30483046 volumes’’, a kind of interface effect, between the core and the shell critically govern the evolution of the components of Young’s modulus for a core–shell nanowire indeed. Secondly, we turn to the stresses arising from the mismatch of lattice constants between the core and the shell. As we know, the lattice constant of bulk Si is smaller by about 4% than that of bulk Ge. For a given core–shell nanowire, the surface atoms of the Ge-core are compres- sively strained by the Si-shell, meanwhile the Si-shell atoms are tensibly strained by the Ge-core. Moreover, the distribution of the strain around the interface somehow correlates with the composition of the core–shell nanowire. These aspects are reflected in the averaged bond lengths of Si–Si, Ge–Ge and Si–Ge varying with the composition as shown in Fig. 7, from which we observe that all of the averaged bond lengths decrease with increasing the composition in Si-core/Ge-shell nanowires, but increase with increasing the composition in Ge-core/Si-shell nano- wires yet. As speculated above, such different strains around the interface also correlate with the variation of Young’s modulus of the core and the shell in a core–shell nanowire. In order to illustrate this relat ion, we recall a simple system consisting of two atoms, in which the two atoms interact with each other. We know that a loaded compressive strain around the equilibrium of the two-atom system makes a larger stress than a loaded tensile strain with the same amplitude. Combining this with formula (2) an d (3), we may conclude that a compressive strain makes a larger increment of Young’s modulus than a tensile strain. Note that the interaction between any two atoms in a concerned nanowire can be similarly described by such a two-atom model. Hence, for our core–shell nanowires, the Ge-cores that are compressed by the connected Si-shells show larger values of Young’s modulus than the corresponded Si-cores, even though Young’s modulus of bulk Ge along h111i direction is lower by about 2 GPa than that of bulk Si along h111i direct ion [24]. The similar situation occurs for the Si-shells and the Ge-shells when x40:35, as shown in Fig. 5. Based on the calculated Y core , Y shell and the fractional volumes, we easily obtain Young’s modulus of the core– shell nanowires with using formula (6), which are plotted in Fig. 2(b). As shown, the dispersion of Young’s modulus of the core–shell nanowires matches that displayed in Fig. 2(a), indicating that the evaluated Young’s modulus by formula (6) is reliable qualitatively. We should point out that (1) the definition of the ‘‘physical volume’’ for the core or the shell in a core–shell nanowire as discussed above does not affect the values of Young’s modulus of the core– shell nanowire; (2) although the trends of the fractional volumes varying with the composi tion are quite similar to those of the ðV core =VÞY core and ðV shell =VÞY shell , the fact that the summation V shell =V + V core =V ¼ 1 does always keep at each composition, whereas the summation ðV core =VÞY core þðV shell =VÞY shell shown in Fig. 2(b) are dependent on the composition implies that the evolution of the Y core and the Y shell with the composition plays a critical role in the composition-dependent trend of Young’s modulus for an entire core–shell nanowire. In addition, as shown in Fig. 5, the trend of Young’s modulus of the considered Si-shells roughly keeps pace with that of the Ge-shells; but Young’s modulus of the Ge-cores show an opposite trend against the Si-cores. Such distinct behavior in Young’s modulus of Ge-cores a nd Si-cores are mainly associated with the different trends of the Young’s modulus for Ge-core/Si-shell and the Si-core/Ge-shell nanowires. 4. Summary In summary, we calculate Young’ s modulus of Ge-core/ Si-shell and Si-core/Ge-shell nanowires systematically. We find that as the composition of the core–shell nanowire ARTICLE IN PRESS 0.0 Composition 2.30 2.35 2.40 2.45 2.50 Bond length (Ang) Si−Si Ge−Ge Si−Ge 0.0 Composition 2.30 2.35 2.40 2.45 2.50 Bond length (Ang) Si−Si Ge−Ge Ge−Si 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Fig. 7. (Color online) Variation of averaged bond lengths of Si–Si, Si–Ge and Ge–Ge in (a) Si-core/Ge-shell and (b) Si-core/Ge-shell nanowires as a function of composition. X.W. Liu et al. / Physica E 40 (2008) 3042–3048 3047 increases, Young’s modulus of Ge-core/Si-shell nanowires increases to a maximal value then drops down, while Young’s modulus of Si-core/Ge-shell is increases. These results are found to be tightly correlated with the mismatch of the lattice constants between Si and Ge at the interface. In addition, the relation between Young’s modulus and the volumes for the core and the shell in a core–shell nanowire is discussed in detail. We point out that the analysis about the basic trends in Young’s modulus of the Si/Ge core– shell nanowires can be helpful for understandings of the mechanical properties for other kinds of core–shell nanowires. Acknowledgments This work is partially supported by the Fund of University of Science and Technology of China, the Fund of Chinese Academy of Science, and by NSFC with code number of 50121202, 60444005, 10574115 and 50721091. B.C. Pan thanks the support of National Basic Research Program of China (2006CB922000). We thank B. Xu, R.L. Zhou and H.Y. He for valuable comments. References [1] L. Zhang, R. Tu, H. Dai, Nano Lett. 6 (2006) 2785. [2] J E. Yang, C B. Jin, C J. Kim, M H. Jo, Nano Lett. 6 (2006) 2679. [3] X. 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B 71 (2005) 125434. [15] C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Phys. Rev. Lett. 96 (2006) 075505. [16] B. Lee, R.E. Rudd, Phys. Rev. B 75 (2007) 195328. [17] B. Lee, R.E. Rudd, Phys. Rev. B 75 (2007) 041305. [18] A.J. Kulkarni, M. Zhou, F.J. Ke, Nanotechnology 16 (2005) 2749. [19] R.Q. Zhang, et al., Chem. Phys. Lett. 364 (2002) 251. [20] F.H. Stillinger, T.A. Weber, Phys. Rev. B 31 (1985) 5262. [21] K. Ding, H.C. Anderson, Phys. Rev. B 34 (1986) 6987. [22] M. Laradji, D.P. Landau, B. Dnweg, Phys. Rev. B 51 (1995) 4894. [23] E.M. Ronald, B.S. Vijay, Nanotechnology 11 (2000) 139. [24] In order to recheck the reliability of this potential, we evaluate the bulk modulus of Si and Ge to be 101.52 and 78.62 GPa, being in excellent agreement to the reported values of 101.0 and 79.0 GPa [K.A. Gschneidner, in Solid State Physics, vol. 16, Academic Press, 1964], and the Young’s modulus of bulk silicon and bulk ger- manium along ½111 direction to be Y Si ½111 ¼ 143:37 GPa, Y Ge ½111 ¼ 141:964 GPa. ARTICLE IN PRESS X.W. Liu et al. / Physica E 40 (2008) 3042–30483048 [...]... in a core–shell nanowire is discussed in detail We point out that the analysis about the basic trends in Young’s modulus of the Si /Ge core– shell nanowires can be helpful for understandings of the mechanical properties for other kinds of core–shell nanowires Acknowledgments This work is partially supported by the Fund of University of Science and Technology of China, the Fund of Chinese Academy of Science,... Physica E 40 (2008) 3042–3048 increases, Young’s modulus of Ge- core /Si- shell nanowires increases to a maximal value then drops down, while Young’s modulus of Si- core /Ge- shell is increases These results are found to be tightly correlated with the mismatch of the lattice constants between Si and Ge at the interface In addition, the relation between Young’s modulus and the volumes for the core and the. .. Stillinger, T.A Weber, Phys Rev B 31 (1985) 5262 [21] K Ding, H.C Anderson, Phys Rev B 34 (1986) 6987 [22] M Laradji, D.P Landau, B Dnweg, Phys Rev B 51 (1995) 4894 [23] E.M Ronald, B.S Vijay, Nanotechnology 11 (2000) 139 [24] In order to recheck the reliability of this potential, we evaluate the bulk modulus of Si and Ge to be 101.52 and 78.62 GPa, being in excellent agreement to the reported values of. .. and 78.62 GPa, being in excellent agreement to the reported values of 101.0 and 79.0 GPa [K.A Gschneidner, in Solid State Physics, vol 16, Academic Press, 1964], and the Young’s modulus of bulk silicon and bulk germanium along ½1 1 1Š direction to be Y Si 1 1Š ¼ 143:37 GPa, Y Ge1 1Š ¼ ½1 ½1 141:964 GPa ... partially supported by the Fund of University of Science and Technology of China, the Fund of Chinese Academy of Science, and by NSFC with code number of 50121202, 60444005, 10574115 and 50721091 B.C Pan thanks the support of National Basic Research Program of China (2006CB922000) We thank B Xu, R.L Zhou and H.Y He for valuable comments References [1] L Zhang, R Tu, H Dai, Nano Lett 6 (2006) 2785 [2] J.-E... 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X.W Si- core /Ge- shell nanowires at the level of density functional theory, where the composition is defined as the ratio of the number of atoms of the core to the number