APPLIED PHYSICS LETTERS 91, 203112 共2007兲
Structural properties and magic structures in hydrogenated finite and infinite silicon nanowires A. D. Zdetsis,a兲 E. N. Koukaras, and C. S. Garoufalis Department of Physics, University of Patras, Patra, GR-26500, Greece
共Received 18 August 2007; accepted 25 October 2007; published online 14 November 2007兲 Unusual effects such as bending and “canting,” related with the stability, have been identified by ab initio real-space calculations for hydrogenated silicon nanowires. We have examined in detail the electronic and structural properties of finite and infinite nanowires as a function of length 共and width兲 and have developed stability and bending rules, demonstrating that “magic” wires do not bend. Reconstructed 2 ⫻ 1 nanowires are practically as stable as the magic ones. Our calculations are in good agreement with the experimental data of Ma et al. 关Science 299, 1874 共2003兲.兴. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2813019兴 One-dimensional nanomaterials and, in particular, thin silicon nanowires 共SiNWs兲, are expected to play a key role in future nanotechnology due to the central role of Si in the semiconductor industry and the existing fabrication technologies. In particular, very thin SiNWs have very important 共electronic and other兲 properties1,2 and promising applications. Ma et al.3 have developed SiNWs with diameters of 1.3– 7 nm, passivated by surface hydrogen, which makes them very stable compared to similarly H-terminated Si wafer surfaces. This difference in stability is attibuted to the different arrangements of SiH2 and SiH3 on the SiNW’s surfaces. For SiH2 phases formed on a flat Si共001兲 wafer, there are generally two equivalent domains with Si–H bonds orthogonal to each other. However, Ma et al.3 found only one domain on the 共001兲 facet of SiNWs with Si–H bonds perpendicular to the axis of the wire. Apparently,3 the bending stress preferentially favors the formation of Si–H bonds perpendicular to the wire’s axis. This important effect, suggested by the experimental data of Ma et al.,3 has not been addressed so far theoretically. Actually, recent theoretical work4 motivated by the work of Ma et al.,3 shows stable SiNWs with both parallel and perpendicular 共to NW axis兲 Si dihydrides. The effect of Si dihydrides in the surface of wafers has been addressed by Northrups5 in his study of Si共100兲H surfaces. Northrup showed that the existence of neighboring SiH2 species lying on the same plane produces a repulsive interaction and induces significant amounts of strain, leading to a rotation of the surface SiH2 groups and a canted-row dihydride structure. In SiNWs, this strain can be relieved through the additional mechanism of bending, as will be verified in the present work, dealing with the influence and consequences of bending stresses on the stability and other properties 共including reconstruction6兲 of SiNWs grown along the 共110兲 direction of bulk Si. To this end, we work in real space 共as opposed to k-space supercell calculations, which cannot allow bending兲 and consider finite length SiNWs, which allows study of length variation, too. We have performed both unconstrained and constrained 共to keep the NWs straight as in supercell calculations兲 density functional calculations and structural optimizations on representative SiNW models based on the STM images of Ma et al.3 The a兲
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generalized gradient approximation functional of Perdew, Burke, and Ernzerhof 共PBE兲 共Ref. 7兲 and the hybrid functional of Becke, Lee, Yang, and Parr 共B3LYP兲 共Ref. 8兲 were used with the SVP 共Ref. 9兲 and TZVP 共Ref. 10兲 basis sets using the program packages TURBOMOLE10 共for finite NWs兲 and CRYSTAL11 共for infinite NWs兲. The convergence criteria in both cases were smaller than 10−4 atomic units for the residual force. The Ahlrichs method12 was used for finite NWs, whereas the k-point sampling of the Brillouin zone was carried out using an 8 ⫻ 8 Pack-Monkhorst net.13 Figure 1 shows the axial view of representative NWs, while the corresponding cross sections are given in Fig. 2. The NW in Fig. 1共a兲 is the simplest and thinnest NW we have examined, containing no parallel dihydrides. As a consequence, it does not bend. However, its stability is considerably lower compared to similar NWs of larger cross section. The NW in Fig. 1共b兲, having parallel dihydrides, is similar to the one given by Chan et al.4 in their Fig. 2.11. For this NW constrained optimization shows no bending, as expected. However, unconstrained optimization leads to bending, as shown in the figure. To evaluate the relative stability of the SiNWs, we use the cohesive energy Ecoh which depends on the NW’s length and width, defined by the relation: Ecoh = 关BE共NW兲 + HNH兴/NSi , where BE共NW兲 is the binding energy of the NW and NSi and NH are the number of Si and H atoms, respectively, with H the chemical potential of H. Doing so, we have effectively removed the energy contribution of all Si–H bonds in every system. This is a common method for the calculation of the cohesive energy.14 With this definition of cohesive energy, we find for the 共constrained兲 straight nanowire of Fig. 1共a兲 Ecoh = 3.44 eV at 100 Å, compared to 3.50 eV for the corresponding bent NW of Fig. 1共b兲. Thus, the cohesive energy of the bent NW, although higher compared to the same unbent NW, is quite lower compared to NWs of the same length without parallel dihydrides, as in Fig. 1共e兲. This is due to residual strains which are not completely relieved by bending. Similarly, we find that all NWs with a closed-structure cross section will not bend if they have no parallel dihydrides, whereas all NWs with parallel dihydrides will bend. This can be considered as nanowire nonbending rule. Obviously such nonbend-
0003-6951/2007/91共20兲/203112/3/$23.00 91, 203112-1 © 2007 American Institute of Physics Downloaded 17 Nov 2007 to 150.140.159.227. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
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Zdetsis, Koukaras, and Garoufalis
FIG. 1. 共Color online兲 Axial view of some representative NWs.
ing NWs are more stable, since strains induced by the repulsion between nearby hydrogens are absent. On the other hand, NWs with parallel dihydrides, which are distributed symmetrically with respect to the NW’s axis, will not bend either due to a balancing out of opposing strains 关Figs. 2共f兲 and 2共j兲兴. However, their stability will be much lower 关3.34 eV 共see Table I兲兴 compared not only to magic but also to nonmagic NW with only one row of parallel dihydrites. Such nonbending but unstable NWs, under the “frustration” of the combined strains, tend to slightly stretch during relaxation. This is also true for the infinite NWs where the stretching is much larger 共Si–Si bonds stretch by more than 0.2– 0.3 Å兲 due to the lack of canting 共which relaxes part of the strains in the finite NWs兲. Up to now, we have examined canting and bending as additional mechanisms of relaxation of NWs, which have not been fully addressed in the theoretical works on thin hydrogenated SiNWs, at least as a result of the relaxation process. The hypothesis of bending stresses made by Ma et al.3 has not been observed in other theoretical calculations, mostly due to the supercell nature of these calculations 共which does not allow for bending兲. In Figs. 1共c兲 and 1共d兲, we show NWs with reconstruction, which is an additional and very effective mode of relaxation.6 The NW of Fig. 1共c兲 was obtained from the NW of Fig. 1共b兲 by replacing the row of parallel dihydrides on the top by a 2 ⫻ 1 reconstruction pattern. Under relaxation this NW bends toward the reconstructed side. The combined effect of reconstruction and bending relieves a very large portion of the strains. As a consequence, the stability of this NW is second to the best by a very small margin, compared to the “magic” NW of Fig. 1共e兲 共the term magic was initially used for such a NW by Chan et al.4兲. The corresponding 3 ⫻ 1 reconstructed NW in Fig. 1共d兲 is less stable 共by 0.02 eV兲. The difference in cohesive energy between the nonmagic 关Fig. 1共f兲兴 and magic 关Fig. 1共e兲兴 NWs is 0.25 eV, substantially larger compared to the difference between other structures. The NW in Fig. 1共f兲 has two symmetric rows of parallel dihydrides resulting in the lowest stabil-
ity, although it does not bend. In contrast, the NW in Fig. 1共g兲 which has two symmetric 2 ⫻ 1 reconstructed surfaces not only does not bend but also its stability is practically as high as the simple 2 ⫻ 1 reconstructed NW in Fig. 1共c兲. The NW in Fig. 1共h兲 has a slightly larger cross section but lower stability because it has three rows of parallel dihydrides 共two on the top and one in the bottom兲. As before, this NW bends away from the 共largest number of兲 dihydrides. The NWs in Figs. 2共i兲 and 2共j兲, which are similar to the NWs in Figs. 2共e兲 and 2共f兲 but with much larger cross section, are more stable 共see Table I兲 as would be expected. In Fig. 3, we plot the cohesive energy as a function of length for various NWs of Figs. 1 and 2. As we can see, Ecoh increases as a function of length 共for a given cross section兲 approaching a saturation value which asymptotically coincides with the value of the infinite NW. This variation of Ecoh implies some type of length variation of bending 共for bending NWs兲. Obviously, the bending has also a dependence on the cross section 共larger diameters implying less bending兲. This is a result of the larger silicon core which tends to retain a bulklike crystalline structure. The surface to volume change and energy partition, which could be expected to be important factors in this length variation, are not so important for the range of lengths examined here 共larger than 0.6 nm兲. The volume-surface ratio in this range of lengths is practically constant and the importance of any edge effects is practically diminished. To examine the length variation of bending, we need to quantify bending. To this end we fit a parabola 共y = ax2兲 and evaluate the coefficient ␣ through a simple nonlinear fit. We find that ␣ increases with increasing length approaching a TABLE I. Cohesive energies of the NWs in Fig. 1, estimated at length of 100 Å. For selected representative SiNWs the cohesive energy at infinite length is given in parenthesis.
NW 共a兲 共b兲 共c兲 共d兲 共e兲
Ecoh 共eV兲 3.38 3.50 共3.51兲 3.59 共3.63兲 3.55 3.60 共3.62兲
NW
Ecoh 共eV兲
共f兲
3.34 共3.37兲
共g兲
3.57
共h兲
3.45 3.80 共3.83兲 3.69 共3.70兲
共i兲 共j兲
FIG. 2. 共Color online兲 Cross section of the corresponding NWs of Fig. 1. Downloaded 17 Nov 2007 to 150.140.159.227. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
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Appl. Phys. Lett. 91, 203112 共2007兲
Zdetsis, Koukaras, and Garoufalis
FIG. 4. 共Color online兲 HOMO-LUMO gap vs length for the NWs of Figs. 1共b兲, 1共e兲, and 1共i兲, based on B3LYP results. The asymptotic values, shown with horizontal lines, correspond to the results for the infinite NW.
FIG. 3. 共Color online兲 Cohesive energy 共in eV兲 as a function of length 共in angstrom兲 for representative SiNWs of Figs. 1 and 2. Dotted horizontal lines correspond to infinite NWs.
constant value. This practically means that the increase of cohesive energy with length is achieved by reducing the strains through increased bending. For the 2 ⫻ 1 reconstructed “near magic” NW of Fig. 1共c兲, the bending is much smaller and almost constant since most of the strain is relieved through reconstruction. The length variation of the highest occupied molecular orbital 共HOMO兲 lowest unoccupied molecular orbital 共LUMO兲 gap, which asymptotically approaches the infinite band gap, is shown in Fig. 4 for three representative NWs. As expected from quantum confinement, the HOMO-LUMO gap decreases as the length increases approaching at infinity the band gap of the infinite NW. Similarly, we can see that the NWs in Figs. 1 and 2共i兲 with the larger diameter have smaller gap than the NWs of Figs. 1 and 2共e兲. Furthermore, the values of the band gaps are in good agreement with the 3.5 eV measured gap of Ma et al.3 at the range of diameters around 1.4 nm. In conclusion, we have developed an alternative criterion of “magicity” in terms of the arrangement of surface dihydrides including bending and reconstruction. We have shown
that the three very effective mechanisms for relief of strains in finite NWs are canting, bending, and reconstruction, with reconstruction being the most efficient one. Our results are in agreement with the work of Ma et al.3 We hope that the effects described here could be properly used in the future in suitable 共devices, hydrogen storage, etc.兲 applications. 1
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