LETTER pubs.acs.org/NanoLett
First-Principles Study of Silicon Nanowire Approaching the Bulk Limit Man-Fai Ng,*,† Michael B. Sullivan,† Shi Wun Tong,‡ and Ping Wu*,† † ‡
Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore Department of Chemistry, National University of Singapore, 3 Science Drive 3, 117543, Singapore
bS Supporting Information ABSTRACT: First-principles density functional theory calculations on hydrogenated silicon nanowires (SiNWs) with diameters up to 7.3 nm are carried out for comparing to experimentally relevant SiNWs and evaluating its radial doping profiles. We show that the direct band gap nature of both the small diameter Æ110æ and Æ100æ SiNWs fades when the diameter reaches beyond about 4 nm, where the difference of direct and indirect band gaps are close, within the experimental measurement uncertainty of (0.1 eV, suggesting the diameter size where the gap nature transition starts. In addition, we reveal that core surface boron (B) codoped SiNW forms more preferably at large diameter than that of the surface surface codoped one, attributing to the lower energy configuration raised by the core B dopant at large diameter SiNW. More importantly, the diameter for such a preferential transition increases as the doping concentration decreases. Our results rationalize photoluminescent measurements and radial doping distributions of SiNWs. KEYWORDS: Silicon nanowire, DFT, band structure, radial doping profile
W
ith the advancement in nanotechnology, silicon nanowires (SiNWs) remain one of the most important nanomaterials since being synthesized successfully more than a decade ago.1 3 Recent device applications utilizing a single nanowire cover a wide range of areas such as transistors for biosensing,4,5 thermoelectric devices,6,7 Li ion battery anodes,8 and solar cells.9 Such nanodevices are not limited to planar designs, but three-dimensional designs have started to appear.4 The experimental successes have resulted in extensive theoretical investigations on SiNWs in recent years. Structure dependent electronic properties, particularly the diameters and growth orientations, are one of the most studied areas for SiNWs within the first-principles formalism.10 20 Such calculations provide accurate and useful information for understanding their material properties at nanoscale. In particular, a recent comprehensive review summarized the structural, electronic, and transport properties of SiNWs.21 From a theoretical point of view, however, the majority of first-principles calculations examine the electronic properties of SiNWs with diameters less than 4 nm, due to the computational difficulty. Going beyond this 4 nm limit with first-principles calculations remains a challenging task to date. However, currently practical sub-10-nm diameter SiNW devices22 24 usually have a channel width larger than about 4 nm. There is a desire to push the theoretical limit to a larger diameter range for the sake of direct comparison with experiment. On the other hand, doping is essential in the fabrication of semiconducting devices. Successful n- and p-type doping on SiNWs have been demonstrated.25 However, the dopants distribution within the nanowire should affect the electronic properties of the nanowire, especially when the nanowire is thin. r 2011 American Chemical Society
Probing the position of an individual atom in a nanostructure experimentally still remains a challenge. Nevertheless, recent progress in controlling the doping in SiNWs has shown that controlled nanoscale doping can be achieved by making use of the crystalline nature of silicon and its self-limiting surface reaction properties.26 Moreover, surface and radial dopant profiles of SiNWs have been revealed by different techniques such as capacitance voltage measurements,27 Kelvin probe force microscopy,28 and Raman spectroscopy.29 31 The generic dopant profiles of SiNWs with diameters of a few tenths of nanometers have been identified: dopants are distributed along the radial direction of the nanowire but unevenly. The uneven distribution along the radial direction can be attributed to the dopant dopant interactions and it is expected to be more significant when the diameters of the SiNWs go down to the sub10-nm regime. Whether the radial dopant dopant interactions can affect the stability of the doped SiNWs and form the preferential distribution remain unclear. Nevertheless, several theoretical studies have reported the radial doping profiles for SiNWs using single dopant with small diameter nanowire.32,33 So far no study has yet been reported on exploring the dopant distribution with multidopant effect along the radial direction in detail. In this Letter, we use first-principles density functional theory (DFT) calculations to investigate the electronic structure hydrogenated SiNWs with diameters up to 7.3 nm and its boron (B) radial doping profiles. We first compare directly, in terms of Received: July 30, 2011 Published: September 26, 2011 4794
dx.doi.org/10.1021/nl2026212 | Nano Lett. 2011, 11, 4794–4799
Nano Letters
LETTER
Figure 1. Cross-section views of Æ110æ and Æ100æ SiNWs with diameters labeled with different colors. A total of 10 and 8 of Æ110æ and Æ100æ SiNWs are used to determine the diameter effect on electronic structures, respectively.
diameter size, the trend of the calculated band gap of SiNWs with experiment.34 We reveal that the direct band gap nature of both the small diameter Æ110æ and Æ100æ SiNWs fades when the diameter reaches beyond about 4 nm. The result compares well with previous photoluminescent measurement experiments of SiNWs.35 On the other hand, we show that there are preferential radial doping profiles for SiNW, depending on the way the surface doping is formed and the doping concentration. Our results provide insights to sub-10-nm diameter SiNWs and predict their properties. Geometry optimizations are performed using DFT within generalized gradient approximation (GGA) in the form of the Perdew Wang exchange-correlation functional36 implemented in the Vienna ab Initio Simulation Package (VASP, version 4.6).37,38 The core valence interaction is described by the projector augmented wave (PAW) pseudopotential method.39 The cutoff energy for the planewave expansion is set at 450 eV. Monkhorst Pack sampling with 1 1 10 (one unit cell) and 1 1 6 (two and three unit cells) k-point grids are used. The SiNW is put in a unit cell with more than 13 Å vacuum spacings in lateral directions to avoid any interactions between the neighboring image nanowires. The SiNWs and the unit cell lattices are fully relaxed until the absolute value of force acting on each atom is less than 0.02 eV/Å. Spin-polarization calculations are used for systems having an odd number of electrons. The thickest Æ110æ and Æ100æ hydrogenated SiNWs with respective diameters of 7.30 and 7.26 nm considered in this work are shown in Figure 1. The cross section of the Æ110æ SiNW is hexagonal containing 2 (100) and 4 (111) surfaces, which is the most stable structure. The Æ100æ SiNW is constructed according to our thinnest Æ100æ SiNW (Si37H28), which has a rectangular cross section with the corner atoms removed. The resulting Æ100æ nanowire contains 4 (100) and 4 (110) surfaces. We increase the diameters of the nanowires in proportion to the number of Si atom. This approach allows a systematic comparison of the electronic properties. The Æ100æ SiNW contains more numbers of Si atoms per unit cell. However, the average number of Si atoms per unit length is similar for these two SiNWs. The thickest SiNW considered in this work contains 1045 Si atoms. The band structures of the Æ110æ and Æ100æ SiNWs with diameters of about 1 and 7 nm are shown in Figure 2a (complete
sets are given in the Supporting Information). Similar to the Æ110æ SiNWs, the Æ100æ SiNWs possess both the direct gap (DG) and indirect gap (IG) except the indirect gap does not appear before the diameter reaching about 2.6 nm. In general, the DG is smaller than the IG but the difference between them diminishes as the diameter increases, which can be explained by the quantum confinement effect. A plot of band gap as a function of diameter is shown in Figure 2b. The curves are fitted according to the effective mass approximation (EMA) equation:40 ESiNW = EBulk + c(1/d)α, where ESiNW and EBulk are the band gaps of the calculated SiNWs and bulk silicon, respectively. d is the diameter of the SiNWs; c and α are fitted parameters. For the experimental data (measured with the Æ112æ SiNWs),34 we fit them using the same equation for comparison. EBulk in this curve is thus the experiment bulk band gap. The fitted α parameters are 1.01 and 1.34 for the Æ110æ and Æ100æ SiNWs, respectively, in agreement with a previous study,18 while other reported α varies from 1.2 to 1.6 for the Æ100æ, Æ110æ, Æ111æ, and Æ112æ SiNWs,12 0.9 for the Æ100æ SiNW and 1.1 for the Æ110æ SiNW,16 and 1.7 for the Æ110æ SiNW.10 The deviations may be attributed to the fact that previous theoretical works considered fewer data points as thinner SiNWs with diameters less than 3 nm were used, while we use thicker SiNWs and have more points to fit. On the other hand, the α parameter of the fitted experimental curve is 1.92 (the Æ112æ SiNW). All these α parameters deviate from EMA where α equals 2. This is an effect of the strong confined states found within small diameter SiNWs. The comparison of the calculated and experimental band gap is shown in Figure 2b. For the growth orientation effect on the band gap of SiNW, it is generally reported that such an effect is negligible when the diameter of the SiNW goes beyond a certain size. We define here that within the regime of 0.1 eV, in conjunction with the usual measurement uncertainty ((0.1 eV) of band gap in experiment, the growth orientation effect is insignificant. In our case, it is the SiNW with a diameter beyond 4 nm. Hence, for the Æ110æ and Æ100æ SiNWs with diameters of 7 nm, the band gap difference is already very small (
