Structural stability, electronic properties, and

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Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. ... The stability of the cagelike nanowires is determined to lie.
PHYSICAL REVIEW B 74, 125311 共2006兲

Structural stability, electronic properties, and quantum conductivity of small-diameter silicon nanowires Inna Ponomareva* Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA

Madhu Menon† Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, Kentucky 40506, USA

Ernst Richter‡ DaimlerChrysler AG FT3/SA, Wilhelm-Runge-Strasse 11, 89081 Ulm, Germany

Antonis N. Andriotis§ Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, Heraklio, Crete, 71110, Greece 共Received 3 April 2006; revised manuscript received 24 May 2006; published 19 September 2006兲 Structures and energetics of various types of silicon nanowires have been investigated using quantum molecular dynamics simulations to determine the most stable forms. The tetrahedral type nanowires oriented in the 具111典 direction are found to be the most stable. The stability of the cagelike nanowires is determined to lie somewhere between this and tetrahedral nanowires oriented in other directions. Furthermore, their electrical conducting properties are found to be better than those of tetrahedral nanowires, suggesting useful molecular electronic applications. DOI: 10.1103/PhysRevB.74.125311

PACS number共s兲: 68.65.⫺k

I. INTRODUCTION

The current interest in silicon nanowires 共Si NWs兲 is mainly derived from their tremendous technological potential in nanoscale devices.1–5 Their sensor applications have also been demonstrated recently.6,7 More recently, experimentalists have succeeded in synthesizing sub 10 nm Si NWs.8–11 This has opened up the possibility of testing confinement effects in these systems. An understanding of their detailed structural properties is, therefore, essential before determining their electronic and sensing properties. The structural predictions for small diameter Si NWs were first made by Menon and Richter12 and Marsen and Sattler.13 Subsequently, theoretical structural determinations of the Si NWs were made by other groups.12–20 Most of these works assign a crystalline core for these nanowires, surrounded by threefold coordinated surface atoms in configurations resembling various well-known surface reconstructions for bulk Si. The electronic properties of Si NWs have also been studied and reveal many intriguing features.12,17,19–21 In particular, for very small diameters, the band gap is determined to be direct.12,21 Their conductivity has also been measured.10,22–25 The theoretical conductivity calculations have revealed the cagelike wires to be better conductors.19 The large surface-to-bulk ratio of the Si NWs can result in their surface reconstruction significantly influencing their energetics, electronic, and conducting properties. While the fourfold coordination of the crystalline bulk of Si NWs has been unequivocally demonstrated in experiments, the precise crystalline structure of the core atoms has not been conclusively confirmed. This is because nanowires of both single crystalline types that include tetrahedral and clathrate or 1098-0121/2006/74共12兲/125311共5兲

cagelike26 as well as polycrystalline types contain fourfold coordinated atoms at the core. In fact, Zhao et al.15 have compared the energetics of formation of Si NWs with singlecrystalline and polycrystalline cores and suggested that for very thin 共1 – 3 nm兲 nanowires, polycrystalline maybe a valid candidate for the lowest energy structure. This was, however, disputed by Ponomareva et al.,19 who, using quantum as well as classical molecular dynamics simulations, have shown single-crystalline tetrahedral type wires grown in the 关111兴 direction to be the most stable and polycrystalline wires to be unstable. Given the fact that the bulk clathrate forms are very close in energy to the tetrahedral form of Si, one could expect that in Si NWs the corresponding energy difference could be even smaller due to the natural surface reconstruction afforded in cagelike forms. Experimentally, Si NWs have been grown in various orientations including the 关111兴, 关110兴, and 关100兴 directions. Indeed, when the diameters get very small, the edge effects become very important.27 This is because the surface reconstruction is further modulated by the curvature effects. This has a profound effect on the energetics and the electronic structure of the Si NWs. In the present work we provide a systematic study of the stability analysis of Si NWs by considering tetrahedral as well as clathrate or cagelike wires. The polycrystalline wires were found to be considerably higher in energy and unstable when subjected to higher temperatures.19 All the nanowires were fully relaxed without any symmetry constraints using the generalized tight-binding molecular dynamics 共GTBMD兲 scheme of Menon and Subbaswamy.28 This scheme has been successfully used in obtaining geometries and vibrational properties of various bulk phases of Si as well as Si clusters of arbitrary sizes.28 All the Si NWs were taken to be of

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FIG. 1. Figure showing a representative set of small diameter Si NWs consisting of 共a兲 tetrahedral 共top兲, 共b兲 Si34-clathrate 共middle兲, and 共c兲 Si46-clathrate 共bottom兲.

infinite length and modeled by large supercells incorporating a constant pressure 共“movable wall”兲 ensemble.29 This allows for a simultaneous relaxation of lattice and basis degrees of freedom. A uniform grid consisting of 126 k-points were used in the calculation of forces. All nanowires had crystalline cores consisting of fourfold coordinated atoms and threefold coordinated surface atoms. Surface reconstructions for all structures were obtained carefully. The GTBMD calculations were supplemented by ab initio calculations for the lowest few diameter nanowires. The ab initio calculations were carried out using the density functional theory 共DFT兲 method as implemented in the Quantum-ESPRESSO package.30 The calculations were carried out using an exchange-correlation functional introduced by Perdew, Burke, and Ernzerhof31 and ultrasoft pseudopotentials. Wave functions were expanded in a plane wave basis to a 340 eV cutoff. A kinetic energy cutoff of 1633 eV was used for the charge density. A 1 ⫻ 1 ⫻ 4 Monkhorst and Pack32 mesh of k-points have been used to sample the Brillouin zone. II. STABILITY OF SI NWS

We investigate the stability of single crystalline Si NWs consisting of tetrahedral and clathrate types with diameters ranging from 1 to 6 nm using the GTBMD scheme. The tetrahedral wires oriented in 具111典, 具110典, and 具100典 directions were considered. The cagelike Si NWs used in this work were carved out from Si clathrate structures consisting of a face centered cubic 共fcc兲 lattice with a 34 atom basis 共Si34-clath兲 as well as a simple cubic 共sc兲 lattice with a 46 atom basis 共Si46-clath兲.26 The bulk Si34-clath structure can be visualized as a three-dimensional 共3D兲 periodic arrangement of Si20 and Si28 cages with shared faces. Similarly, the bulk Si46-clath structure can be visualized as a 3D periodic arrangement of Si24 cages with shared faces interpenetrated by Si20 cages. The interstitial spaces are occupied by Si2 dimers. Both these clathrate forms are known to be the most stable clathrate forms for bulk Si. The cohesive energies for the bulk tetrahedral, Si34-clath and Si46-clath obtained using the GTBMD scheme are, respectively, 5.27, 5.24, and 5.23 eV.

FIG. 2. 共Color online兲 Plot showing the diameter dependence of the energy for the Si NWs obtained using the GTBMD scheme for the fully relaxed nanowires. The zero of energy is taken to be the cohesive energy of bulk tetrahedral Si 共5.27 eV兲 obtained using the GTBMD scheme. The inset shows the corresponding dependence obtained using the ab initio method. The zero of energy here is taken to be the bulk cohesive energy using the ab initio method 共4.59 eV兲. Both methods show the tetrahedral wires oriented in the 具111典 direction to be the most stable.

The energetic ordering of these structures is identical to that obtained using first principles methods.31 In Fig. 1 we show a few representatives for tetrahedral and clathrate type Si NWs for very narrow diameters. It should be noted, however, that all the bare surface reconstruction considered in the present work is different from those seen in experiments where the surface of the Si NWs are passivated. Nevertheless, the properties depending primarily on the underlying bulk crystalline structure should be comparable. In Fig. 2, we show the diameter dependence of the cohesive energy for the Si NWs obtained using the GTBMD scheme for the fully relaxed nanowires of diameter up to 6 nm. As seen in the figures, the tetrahedral wires oriented in the 具111典 direction are found to be the most stable. This finding is supported by the ab initio results for very small diameter Si NWs shown in the inset of the same graph. Interestingly, the Si34-clath and Si46-clath wires have lower energies than the tetrahedral wires oriented in the 具110典 and 具100典 directions. Even though the bond angles for the core atoms in clathrate nanowires deviate from the ideal tetrahedral case, the cage closure provides a natural surface reconstruction with minimal strain even at very small diameters. This makes the clathrate nanowires energetically competitive with the tetrahedral nanowires at small diameters. III. ELECTRONIC PROPERTIES

We next investigate the size and orientation dependence of the band gap of tetrahedral Si NWs up to 6 nm in diameter. As stated before, due to the small diameters, the surface reconstruction will be affected not only by the surface effects, but also the curvature effects.27 The electronic structure, in turn, will sensitively depend on both these two fac-

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FIG. 3. 共Color online兲 Plot showing the dependence of the gap on the diameter for the tetrahedral Si NWs. The filled circles indicate direct band gaps, while the unfilled circles indicate indirect gaps. The gap value near the 5 nm diameter for 具111典 and 具100典 Si NWs is very close to the bulk gap value for Si in the diamond phase.

tors. All the structures were fully relaxed without any symmetry constraints and their surface relaxation carefully obtained. For band structure calculations, we use a five orbital sp3s* tight-binding Hamiltonian33 which ensures an accurate conduction band edge. This model accurately reproduces the band gaps in bulk Si.33 The exciton energies for Si nanocrystals obtained using this model are in quantitative agreement with photoluminescence data.34 Furthermore, the model also shows agreement with the correlation between visible and infrared photoluminescence for porous Si.35 Figure 3 shows a plot of band gap as a function of the diameter for the three orientations of the nanowires. As seen in the figure, there is a great degree of anisotropy in the gap values. The band structure analysis reveals further anisotropy in the direct to indirect transition as the diameter increases. The filled circles in Fig. 3 indicate direct band gaps, while the unfilled circles indicate indirect gaps. For the 具111典, 具110典, and 具100典 wires, the direct-indirect transition occurs between the diameters 4.5–5.3, 1.39–2.10, and 1.44– 2.21 nm, respectively. The gap values are the smallest for the 具110典 wires. This is in agreement with other theoretical works.36 Another unique aspect of the 具110典 wire is that

FIG. 5. 共Color online兲 Current vs voltage 共I-V兲 characteristics for the single crystalline nanowires of tetrahedral and cagelike geometries obtained under symmetric bias. Both cagelike crystalline nanowires show better conducting properties than the tetrahedral nanowire under low bias.

the direct band gap is at the zone boundary, whereas for the 具111典 and 具100典 wires, the direct band gaps are at the zone center 共⌫-point兲. Our results suggest the 具111典 wires to be better candidates for studying quantum confinement effects since the direct band gap persists all the way up to 4.5 nm. In Fig. 4 we show the electronic band structures for two 具111典 nanowires, one small 共diameter= 1.44 nm, left兲 and one large 共diameter = 5.30 nm, right兲 showing direct and indirect band gaps, respectively. IV. QUANTUM CONDUCTIVITY

We investigate the quantum conductivity of various nanowires next. In Fig. 5 we present the calculated I-V curves for the tetrahedral and cagelike geometries obtained under symmetric bias. All nanowires have similar diameters 共⬇2 nm兲 and finite lengths 共⬇4 nm兲. The conductivity is calculated using a formalism incorporating the transfer Hamiltonian approximation 共THA兲 method 共see, for example, Ref. 37 and references therein兲, according to which the applied voltage is assumed to be changing linearly from one end of the nanowire to the other. For each value of the bias voltage the

FIG. 4. Band structures for 具111典 Si NWs of diameters 1.44 nm 共left兲 and 5.3 nm 共right兲. The dashed lines indicate the Fermi levels. The smaller diameter nanowire has direct band gap as a result of the quantum confinement, while the larger diameter nanowire has indirect band gap with a value close to bulk Si in the diamond phase. The inset shows a magnified shot of the lowest conduction bands.

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Fermi energy of the system is calculated, thereby achieving a zeroth order of self-consistency. The Hamiltonian used in the THA method is the same as used in performing the GTBMD structural relaxation for the Si NWs. This allows for consistency in our calculations. The I-V behavior is identical for the different orientations of the tetrahedral nanowires, indicating that conducting channels are not affected by the various surface reconstructions but by the underlying bulk crystalline structure. Interestingly, the cagelike nanowires, and in particular, the Si46 clathrate nanowires, have higher conductivity under low bias voltage when compared to the tetrahedral nanowires. This may have important consequences in molecular electronic uses for the Si NWs. One of the I-V features found is the transport gap 共⬎2 eV兲. It should be noted that such a gap disappears in doped nanowires which exhibit a rather ohmic type behavior.23,25 We investigate the effect of hydrogen adsorption on the conducting properties next. In Fig. 6 we present the calculated I-V curves for bare as well as 50% and 100% H-covered Si NWs of diameter 1.44 nm oriented in the 具111典 direction under symmetric bias. The wires are of finite length 共⬇9 nm兲 and the Si atoms at both ends were passivated with hydrogen. The Hamiltonian used in the THA method is the GTBMD Hamiltonian containing hydrogen parametrization.37 All structures have been fully relaxed with the GTBMD method. As seen in Fig. 6, the current values are similar for bare and 50% H-covered Si NWs, but drop to almost zero for 100% H-coverage for the same voltage range. This can be understood by an examination of the DOS 共see inset to Fig. 6兲. The Fermi energies, EF’s, are indicated by dashed lines. At 100% H-coverage, there is a complete removal of all the gap states and a considerable widening of the gap. The I-V curves in Fig. 6 are qualitatively similar to the experimental I-V curves for Si NWs reported in Ref. 10. The H-induced changes in the electronic band gaps and transport properties are significant in that they can be detected in an external circuit for chemical, gas, and/or biosensing applications.

In summary, we have presented results for stability, electronic structure, and conductivity for different Si NWs. Our results show that tetrahedral Si NWs oriented in the 具111典 direction are the most stable and best suited for investigating quantum confinement effects. Our results also show that the cagelike nanowires have higher electrical conductivity and are better suited for molecular device applications than the tetrahedral nanowires. ACKNOWLEDGMENTS

The present work was supported through grants by NSF 共ITR-0221916兲, DOE 共DE-FG02-00ER45817兲, and USARO 共W911NF-05-1-0372兲.

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*Electronic address: [email protected]

FIG. 6. 共Color online兲 Current vs voltage characteristics for bare as well as 50% and 100% H-covered Si NWs of diameter 1.44 nm and finite length oriented in the 具111典 direction. The inset shows the corresponding DOS for the Si NWs. There is a complete removal of all gap states on 100% H-saturation, resulting in a significant drop in the current value for the voltage range shown.

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