Revising morphology of 〈111〉-oriented silicon and germanium nanowires
- Dmitri B Migas^{1}Email author,
- Victor E Borisenko^{1},
- Rusli^{2} and
- Cesare Soci^{3}
https://doi.org/10.1186/s40580-015-0044-8
© Migas et al.; licensee Springer. 2015
Received: 29 December 2014
Accepted: 20 January 2015
Published: 10 August 2015
Abstract
By means of ab initio calculations we show that morphology of 〈111〉-oriented silicon and germanium nanowires is defined by {112} and {011} facets. Changes in nanowire morphology are predicted to involve a partial transformation of {011} facets in favor of {112} facets even though the latter ones act as edges between adjacent {011} facets. Our estimates of surface energies clearly indicate a (112) surface to be thermodynamically preferable with respect to a (011) surface for both silicon and germanium. These findings can explain experimental observations of {112} facets in round-like and triangle-like morphologies of 〈111〉-oriented silicon nanowires.
Keywords
1 Background
Nowadays silicon and germanium nanowires (SiNWs and GeNWs) are considered to be promising and, at the same time, accessible building blocks for various applications at nanoscale [1]. In fact, SiNWs and GeNWs having mostly 〈011〉, 〈111〉 and 〈112〉 orientations can easily be grown by different methods involving the vapor-liquid-solid growth mechanism [1-3]. Experimental data also indicate a clear faceting nature of morphology of these nanostructures [3,4] while theoretical calculations of the total energy of SiNWs show morphology, which is characterized by various facets, to be one of the key parameters to define the growth orientation especially at small diameters of the NWs when the surface energy is comparable to or dominates over the volume energy [5].
It is also widely accepted that the Wulff construction can predict morphology of a NW on the basis of precise information on surface energies of different surfaces. In the case of silicon the most thermodynamically stable surfaces according to experimental observations are (111), (001), (113) and (011) [6], that is also supported by results of ab initio calculations [7,8]. For germanium one can expect the same issue because of the same nature of chemical bonding in silicon and germanium and of comparable differences in surface energies for various surfaces with respect to silicon as theoretically predicted [7,8]. In addition to these low index surfaces, some other surfaces such as Ge(105) with the relatively low surface energy [9] should also be taken into account. This surface in the case of the proper surface reconstruction is found to be thermodynamically stable and it appears on facets of germanium pyramids formed during heteroepitaxial growth of Ge on Si(001) [10]. Thus, morphology of 〈111〉-oriented SiNWs and GeNWs as suggested by the Wulff construction could be characterized by {011} facets. It is also possible to use {112} facets which play the role of edges between adjacent {011} facets in order to avoid an appearance of surface atoms with two dangling bonds [5]. However, there is some experimental evidence that mainly {112} facets define morphology of SiNWs with 〈111〉 axes and with diameters in the range of 50 – 100 nm [11-16]. The latter fact obviously contradicts the common assumption that the Si(112) and Ge(112) surfaces are less stable (or unstable at all) than any of the (111), (001), (113) and (011) ones. There is one paper [17] where Si(112) surface with the 1 ×1 and 2 ×1 reconstructions has been investigated by first principles techniques indicating the rebonded 1 ×1 reconstruction to be thermodynamically stable even though the difference in surface energy between these two surface reconstructions was marginal. In addition, Si(112) surface was shown to be quite competitive in surface energy with respect to (100) and (111) ones. However, the slab thickness in these calculations was about 0.5 nm and an interaction between opposite surfaces in such a slab could not be excluded. The latter fact can affect values of calculated surface energy. Moreover, another estimates of surface energies for the Si(112) and Ge(112) surfaces with respect to the other surfaces have been done by the modified embedded atom method [18]. However, in such calculations neither structural optimization nor surface reconstruction has been performed [18]. Thus, the predicted relative stability of Si(112) and Ge(112) in Refs. 17 and 18 remains questionable and the appearance of the {112} facets in morphology of SiNWs is not fully understood.
In this paper we present results demonstrating stability of 〈111〉-oriented SiNWs and GeNWs with different morphology obtained by means of the total energy projector-augmented wave method and provide theoretical evidence that {112} facets are really thermodynamically stable and can define the shape of these nanostructures.
2 Methods
The structural optimization of SiNWs and GeNWs has been performed by utilizing the first principles total energy projector-augmented wave method (code VASP) described in detail elsewhere [19-21]. Exchange and correlation potentials were included using the generalized gradient approximation of Perdew-Burke-Ernzerhof [22] in the case of SiNWs and the local density approximation of Ceperly and Alder by the parameterization of Perdew and Zunger [23] for GeNWs. We have applied both the generalized gradient approximation and the local density approximation because the former describes better ground state properties for silicon while the latter fits better germanium. Since no direct comparison between values calculated for silicon and germanium surfaces and/or NWs is planned, our choice of the two approximations looks reasonable. We have considered 〈111〉-oriented SiNWs and GeNWs with different cross sections and diameters, while periodic boundary conditions have been applied along the NW axis with the unit cell parameter (a _{∥}). In order to provide a negligible interaction between neighboring NWs at least 7 Å of vacuum were introduced. The futher increasing in the vacuum thickness did not noticeably affect the total energy. All atoms in SiNWs and GeNWs were allowed to relax. We set the energy cutoff at 300 eV for SiNWs and at 225 eV for GeNWs. The grid of 1 ×1×6 Monkhorst-Park points was used in calculations. Atomic relaxation was stopped when forces on the atoms were smaller than 0.04 eV/Å. To assure the convergence, the final iterations have been performed on the 1 ×1×10 grid. The optimization of a _{∥} was done by gradually increasing/decreasing its value along with the relaxation of the atomic positions till the equilibrium was reached. The bulk lattice parameters (a _{ Si } and a _{ Ge }) were found to be 5.467 Å and 5.646 Å, respectively. The initial a _{∥} was set at \(\sqrt 3 a_{\textit {Si}}\) for SiNWs or at \(\sqrt 3 a_{\textit {Ge}}\) for GeNWs.
In order to calculate surface energies of the (112), (011), (001) and (111) silicon and germanium surfaces we have considered periodic arrangement of slabs separated by 7 Å of vacuum as in the case of NWs. Each slab had thickness of about 4 nm and it was characterized by two equal surfaces. Such a thickness was enough to assure convergence in surface energy with respect to the slab thickness. All of the atoms in the slab were allowed to relax. We adopted the same exchange and correlation potentials and energy cut-off as for SiNWs and GeNWs. The convergence in surface energy was found to be satisfactory (less than 0.01 eV/Å^{2}) on the grid of 9 ×9×1 Monkhorst-Pack points for the (112), (011) and (001) surfaces and on the grid of 7 ×15×1 Monkhorst-Pack points for the (111) surfaces. Atomic relaxation was stopped when the forces on atoms were less than 0.01eV/Å. The surface energy is calculated as a difference between the energy of a silicon or germanium atom in the bulk multiplied by the number of atoms in a slab and the total energy of a slab. Then this difference can be expressed per unit cell or square unit area. In order to calculate a dependence of the surface energy on an in-plane lattice parameter, we have eliminated any residual elastic effect caused by compression or expansion of an in-plane lattice parameter for corresponding bulk cases, where the tetragonal-like distortion has occurred in the unit cell to construct a slab along with relaxation of the lattice parameter which is perpendicular to the corresponding surface plane. This approach has been successfully applied to Ge(105) [9].
3 Results and discussion
3.1 Morphology of SiNWs and GeNWs
The diameter ( d , nm), the number of atoms in the unit cell ( N ) and the lattice parameter along the wire axis ( a _{ ∥ } , Å) for SiNWs and GeNWs with different morphologies after structural optimization
Si{011} | Si{112} | Si{112}-t | |||||||
---|---|---|---|---|---|---|---|---|---|
d | 1.5 | 2.1 | 2.5 | 3.0 | 3.4 | 1.9 | 2.7 | 3.1 | 3.1 |
N | 110 | 170 | 242 | 326 | 422 | 146 | 266 | 362 | 338 |
a _{∥} | 5.524 | 5.483 | 5.473 | 5.470 | 5.469 | 5.432 | 5.486 | 5.470 | 5.480 |
Ge{011} | Ge{112} | Ge{112}-t | |||||||
d | 1.6 | 2.2 | 2.6 | 3.1 | 3.5 | 2.0 | 2.8 | 3.2 | 3.2 |
N | 110 | 170 | 242 | 326 | 422 | 146 | 266 | 362 | 338 |
a _{∥} | 5.429 | 5.513 | 5.562 | 5.588 | 5.607 | 5.292 | 5.552 | 5.581 | 5.579 |
3.2 Stability of SiNWs and GeNWs
In general, the corresponding dependencies for GeNWs (Figure 4, the bottom panel) are similar to the ones of SiNWs. Nevertheless, GeNWs with the nonreconstructed {112} facets are predicted to be thermodynamically competitive. This issue can stem from the fact that such a GeNW is characterized by a _{∥}, which deviates by 6.3 % from the bulk value, providing in turn a significant lowering of the surface energy even at the expense of the bulk energy. In addition, the triangle-like morphology (Figure 3) is not expected for GeNWs having small diameters. This morphology can appear at larger diameters of a NW, when the volume energy dominates over the surface energy, and/or it is mainly defined by the interface energy between a catalytic particle and a NW.
Surface energies (meV/Å ^{ 2 } ) of the (112), (011), (001) and (111) surfaces of silicon and germanium
Si | Ge | |
---|---|---|
(112) with 2 ×1 reconstruction | 89.81 | 72.41 |
(112) with 1 ×1 reconstruction | 92.27 | 74.74 |
(011) | 96.29 | 76.76 |
(001) | 79.77 | 66.91 |
(111) | 83.02 | 71.58 |
4 Conclusions
The results of our ab initio calculations confirm experimental observations that morphology of 〈111〉-oriented SiNWs is characterized by the {011} and {112} facets where the latter ones are predicted to define their shape if diameter of a NW is larger than 2.5 nm. The same behavior has been also observed for GeNWs. The reason of the appearance of the {112} facets in morphology of SiNWs and GeNWs is the lower surface energy of the (112) surface with respect to the (011) surface for both silicon and germanium. Thus, the common assumption that the (112) surface of silicon and germanium is unstable does not hold. Consequently, the appearance of the {112} facets which define morphology of 〈111〉-oriented SiNWs and GeNWs should not come as a surprise. Moreover, the triangle-like morphology with alternate large and small {112} facets of SiNWs is found to be thermodynamically competitive even for NWs with diameters starting from 3 nm. Stability of SiNWs and GeNWs in 〈111〉 growth directions is also improved by the transformation of a portion of the {011} facets into the {112} ones. Further lowering in the total energy can be achieved for SiNWs and GeNWs by forming the sawtooth faceting with {111} and {113} facets instead of the {112} facets as experimentally observed [11-16]. The latter is only possible for NWs with diameters starting from 100 nm because there is enough space to involve facets which are not perpendicular to the plane of the NW cross section and the triangle-like morphology is even more favorable for this purpose. Moreover, the Wulff construction is shown to be reliable to generate initial structures of SiNWs and GeNWs. Finally, our findings can be also useful in investigating of morphology of III-V NWs, such as GaAs, where many of the described features have been experimentally observed [25].
Declarations
Acknowledgements
DBM and VEB thank the Belarusian National Research Program “Convergence” and Belarusian Republican Foundation for Fundamental Research under (Grant No. F14U-001) and Singapore Ministry of Education (project reference MOE2013-T2-1-044) for financial support. The authors are grateful to Filonov AB and Shaposhnikov VL for fruitful discussion and useful suggestions on the results of the paper. This work is supported by the Agreement on Cooperation between Belarusian State University of Informatics and Radioelectronics and Nanyang Technological University.
Authors’ Affiliations
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