NANO LETTERS
Origin of Diameter-Dependent Growth Direction of Silicon Nanowires
2006 Vol. 6, No. 7 1552-1555
C. X. Wang,* Masahiro Hirano, and Hideo Hosono Frontier CollaboratiVe Research, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 226-8503, Japan Received January 16, 2006; Revised Manuscript Received April 25, 2006
ABSTRACT A nucleation thermodynamic model was developed to clarify the diameter-dependent crystallographic orientation of silicon nanowires (SiNWs) grown via the vapor−liquid−solid (VLS) mechanism with an Au catalyst. The calculated critical energies (Er*) and corresponding critical radii (r*) of the SiNWs with 〈111〉 and 〈110〉 orientations as a function of Au-catalyst size (DAu) revealed that the 〈110〉-oriented SiNW with r* is preferred below DAu ) ∼25 nm, but the preferred direction changes to 〈111〉 above DAu ) ∼25 nm. The model indicated that the nucleated SiNW with a radius (r) above r* is stable and continues to grow until the diameter becomes equal to DAu but that the crystallographic orientation is maintained. Thus, the predicted growth direction of the final SiNW with a size of DAu is 〈110〉 for DAu < ∼25 nm and 〈111〉 for DAu > ∼25 nm, which is in excellent agreement with reported experimental results.
Semiconductor nanowires (SNWs) have attracted much attention in the past few years due to their unique fundamental electronic and structural properties, which are suitable for a variety of nanoscale devices, including biologic sensors and electronic and optoelectronic devices.1,2 Among the various types of SNWs, silicon nanowires (SiNWs) are the most attractive due to the central role of Si in current integrated circuit technology.3 In addition, SiNWs exhibit a desirable property for integrated device applications because a single material would form the active layers of nanosized devices as well as the interconnects between devices.4,5 In fact, remarkable progress has recently been reported for the growth of SiNWs and applications to electronic devices such as diodes,2 field-effect transistors,6 logic gates,7 single electron transistors,8 and sensors.9 In particular, the growth direction of a SiNW is diameter-dependent; Wagner and Ellis grew SiNWs using chemical vapor deposition (CVD) and a gold catalyst in the early 1960s and found that SiNWs with diameters greater than 100 nm tended to grow in the 〈111〉 direction.10 Since then, vast knowledge with respect to the growth of SiNWs has been accumulated. It has been clarified that smaller-diameter nanowires primarily grow along the 〈110〉 direction, but larger ones grow along the 〈111〉 direction.11-15 More specifically, Schmidt and co-workers have demonstrated that epitaxially grown SiNWs with diameters greater than 40 nm prefer the 〈111〉 direction, although wires with diameters of less than 20 nm are mostly 〈110〉-oriented. They have also quantitatively explained the observed diameter-dependent orientation based on the free * To whom correspondence should be addressed. E-mail: wangcx@ lucid.msl.titech.ac.jp. 10.1021/nl060096g CCC: $33.50 Published on Web 06/06/2006
© 2006 American Chemical Society
energy of the grown SiNW with a final diameter (TD model).13 On the other hand, one of the present authors (C.X.W.) has proposed a nucleation thermodynamic model (NTD model)16 in which only a nucleated SiNW with a radius larger than the critical radius (r*) is stable and continues to grow to the final diameter. The predicted minimum radius of a SiNW provided by r* in this model agrees well with the experimental findings. However, the NTD model predicts that the crystallographic orientation is highly independent of the SiNW diameter, which contradicts experimental observations. This disparity may be because the NTD model determines the growth direction during the nucleation stage, which is seemingly independent of the final SiNW size, and the crystallographic orientation of the SiNW is nearly constant during the lateral growth of a SiNW to its final size. In this letter, we have improved our NTD model by considering the orientation-dependent surface and interface free energies to clarify the diameter-dependent orientation of SiNWs grown through the VLS mechanism with an Au catalyst. The quantitative relationship between the critical energies for the different growth directions (〈110〉 and 〈111〉) and catalyst diameter (DAu) at the nucleation temperature, which is calculated using the model, reveals that the crystallographic orientation of the nucleated SiNW with a radius of r* changes from the 〈110〉 direction to the 〈111〉 direction at a crossover size of DAu ) ∼25 nm. Thus, the crystallographic orientation of the final SiNW with a diameter of DAu depends on DAu, indicating that the NTD model for VLS growth of SiNW is valid. Thermodynamically, the phase transformation is usually promoted by the difference in Gibbs free energy (∆G).17-19
corresponding interface areas (see Figure 1) given by S1 ) 2πR′h and S1 ) 2πrH. For an incoherent interface between the SiNW cluster and the Au-catalyst liquid droplet, which do not have lattice matching at the interface, σlSi is denoted as σlSi ) (σlv + σSiv)/2.17-19 V is the volume of an Si cluster provided by V ) 1/6πH(H2 + 3L2) - 1/6πh(h2 + 3L2). h, H, and L2 are, respectively, given by h ) R′ -
xR′2 - r2 + r2(R′ cos θ - r)2/(R′2 + r2 - 2R′r cos θ) H ) r 1 - (R′ cos θ - r)/xR′2 + r2 - 2R′r cos θ and L2 ) r21 - (R′ cos θ - r)2/(R′2 + r2 - 2R′r cos θ)
Figure 1. (a) Schematic drawing of SiNWs nucleated on an Au catalyst, which shows the geometrical parameters in eq 2. (b) Nucleation and growth process by the VLS mechanism with an Au catalyst: (1) in the early stage of nucleation; (2) the size of nucleus reaches the critical value, arrows show the growth direction of a SiNW critical nucleus; (3) final SiNW. Arrowheads define the growth direction of a SiNW, demonstrating that the orientation of a SiNW remains constant between the final and the nucleus stages.
Thus, among the different oriented Si clusters that can coexist during the nucleation of a SiNW, only a cluster with minimal free energy in terms of the crystallographic orientation is stable. The others are meta-stable and are remelted or transformed into clusters with a stable orientation in the NTD model. Figure 1a schematically shows the geometrical configuration of the SiNWs nucleation and growth by the VLS mechanism. Here, we ignored the influence of the substrate because the effect of the interface energy between the substrate and the SiNW on the growth direction is much smaller than that of the surface energy (S2) and the interface energy (S1) between the catalyst and SiNW.13,20,21 If (i) both the Au liquid droplet and the Si cluster are perfect spheres, which causes the interface between the droplet and the cluster of a SiNW to be a spherical surface, and (ii) the Si nuclei are mutually noninteractive, then ∆G of a Si cluster formed via the VLS mechanism is expressed as the sum of the interface energy between the Au catalyst and the Si cluster (∆G1), the surface energy (∆G2), and the volume energy (∆G3).17,18 On the basis of the nucleation thermodynamic ∆G ) ∆G1 + ∆G2 + ∆G3
(1)
theory with the help of Figure 1, ∆G1, ∆G2, and ∆G3 can be expressed as (σlSi - σlv)Sl, σSivS2, and ∆gvV, respectively, where σlSi, σlv, and σSiv are the interface energy densities for the liquid droplet/SiNW nuclei, the Au liquid droplet/vapor, and the SiNW nuclei/vapor, respectively. S1 and S2 are the Nano Lett., Vol. 6, No. 7, 2006
where R′ ) DAu/2, r denotes the curvature radius of the liquid droplet and the cluster, and θ is the contact angle between the cluster and the liquid droplet. Meanwhile, based on the Laplace equation and the expression for σlSi, cos θ ) (σlv σSiv)/2σSiv.18 ∆gv is the difference in Gibbs free energy per unit volume expressed by -RT/Vm ln(C/Ceq)22 where R, T, Vm, C, and Ceq are the gas constant, absolute temperature, the mole volume of SiNW, the silicon concentration on the solid, and liquid line of the Au-Si phase diagram, respectively. Thus, eq 2 is obtained from eq 1 for ∆G of the nucleated SiNW. ∆G ) πR′(σlSi - σlv) ×
(
R′ -
x
(
2πσSivr2 1 -
((
r2(R′ cos θ - r)2
R′2 - r2 +
R′2 + r2 - 2R′r cos θ
xR′
(
+
r2(R′ cos θ - r)2
)
+ r2 - 2R′r cos θ
( ))
RT C 1 ln π 6 Vm Ceq
R′ -
(
x x
R′ - r +
3r2 R′ -
2
( ( (
r 1-
×
R′2 + r2 - 2R′r cos θ
R′2 - r2 +
1-
3
2
R′2 + r2 - 2R′r cos θ
R′2 + r2 - 2R′r cos θ
)
R′ cos θ - r
xR′2 + r2 - 2R′r cos θ xR′
(
)
×
)
3
+ r2 - 2R′r cos θ 1-
+
-
R′ cos θ - r 2
3
r2(R′ cos θ - r)2
(R′ cos θ - r)2
3r3 1 -
+
)
R′ cos θ - r 2
)
-
)
×
(R′ cos θ - r)2 R′2 + r2 - 2R′r cos θ
))
(2)
Equation 2 relates the critical energy and DAu at a given 1553
Figure 3. Critical energy of both 〈110〉- and 〈111〉-oriented SiNWs as a function of the size of an Au liquid droplet, which serves as the catalyst for the SiNW growth.
Figure 2. (a) Gibbs free energy of a 〈110〉-oriented Si cluster as a function of the cluster radius for three different Au-catalyst sizes at the nucleation temperature (673 K), R′ ) DAu/2. (b) Gibbs free energy of a 〈111〉-oriented Si cluster as a function of cluster radius for three different Au-catalyst sizes at the nucleation temperature (673 K), R′ ) DAu/2.
surface energy density and critical nuclei radius. In the ∆G2 calculation, a hexagonal column approximates the hemispherical shape of the Si cluster. Then, the surface energy density of the Si cluster (σSiV, the surface energy density of S2 in Figure 1) is equal to the average energy density of the six side surface planes and a top surface plane of the hexagonal SiNW; the surface energy density of the 〈111〉oriented Si cluster is equal to the average surface energy density of the six {110} side surface planes and a {111} top surface plane, but that of the 〈110〉-oriented SiNW is the sum of the four {111} side planes, two {100} side surface planes, and a top {110} surface plane.3,13 On the basis of the approximation, the surface energy densities of the 〈110〉and 〈111〉-oriented Si clusters are 1.36 and 1.4 J/m2, respectively, when the surface energy densities are 1.25 J/m2 for {111}, 1.53 J/m2 for {110}, and 1.28 J/m for the average energy density of six side surfaces of a 〈110〉-oriented SiNW.2,13 We also employed a liquid droplet-gas surface energy density of σlv ) 0.85 J/m2 (ref 23) and a mole volume of a single-crystal silicon of Vm ) 1.2 × 10-5 kg/m3. In addition, based on the Au-Si phase diagram24 and C ∼ 1 (ref 22), eq 2 gives ln C/Ceq ≈ 1.449. The Gibbs free energies at the nucleation temperature (673 K) are calculated as a function of the radius for the 〈110〉- and 〈111〉-oriented Si clusters (Figure 2). The critical radius (r*) of the Si cluster, which is defined as the radius corresponding to ∂∆G(r)/∂r, is 4.0 nm for the 〈110〉-oriented Si cluster and 4.1 nm for the 〈111〉-oriented one. The r* value is nearly independent of DAu. On the other hand, as shown in Figure 3, the critical energy (Er*) or the free energy of the cluster with r* 1554
decreases as DAu increases at a fixed temperature and growth direction, suggesting that the activation energy for the growth of a SiNW (the nucleation barrier of a SiNW) increases as DAu decreases. This finding is consistent with Lieber’s prediction,12 which states that small-sized SiNWs have larger activity energies for growth than large-sized SiNWs. Our finding is also consistent with Kikkawa’s experimental results, which show that SiNWs with a smaller diameter grow slower than those with a larger diameter.25 Figure 3 further demonstrates that the 〈110〉-oriented SiNWs are stable for DAu smaller than ∼25 nm and the 〈111〉oriented SiNWs are energetically more favorable for DAu larger than ∼25 nm, which agrees with the reported experimental observations.11-15 This excellent agreement strongly suggests that the crossover size of the Au catalyst, which is where the growth direction transition may occur, is dominantly governed by the surface and the interface energies of the SiNWs. The good reproducibility of the reported crossover size implies that extrinsic factors, including impurities in the SiNWs and the concentration of the vapor source materials, have a negligible influence on the crossover size. In summary, to obtain clear insight into the origin of the diameter-dependent growth direction of SiNWs using VLS growth and a Au catalyst, we have developed a nucleation thermodynamic model by considering the orientation-dependent surface and interface energies of SiNWs. Through the use of the improved model, the Au-catalyst size dependence of the critical energies for 〈110〉- and 〈111〉oriented SiNWs as a function of Au-catalyst size is analyzed. Our results indicate that the growth direction of a SiNW changes from the 〈110〉-oriented direction to the 〈111〉oriented direction at a critical size of ∼25 nm. The good agreement between the improved NTD model and experimental observations indicates that this NTD model is more appropriate for the growth process of a SiNW via the VLS mechanism with a Au catalyst. Furthermore, because Cai et al. have recently reported that the growth direction of a ZnSe nanowire, which has a similar growth mechanism to SiNWs, is determined by the size of the Au-Ga binary alloy catalyst,20,21 the present approach may not be restricted to SiNWs but may be applicable to nanowire formation of various semiconductor materials. Nano Lett., Vol. 6, No. 7, 2006
Acknowledgment. This work was supported by the Japan Society for the Promotion of Science (JSPS). References (1) AdV. Mater. 2003, 15, Special Issue: One-Dimensional Nanostructures. (2) Cui, Y.; Lieber, C. M. Science 2001, 291, 851. (3) Ma, D. D. D.; Lee, C. S.; Au, F. C. K.; Tong, S. Y.; Lee, S. T. Science 2003, 299, 1874. (4) Lew, K. K.; Redwing, J. M. J. Cryst. Growth 2003, 254, 14. (5) Ross, F. M.; Tersoff, J.; Reuter, M. C. Phys. ReV. Lett. 2005, 95, 146104. (6) Cui, Y.; Zhong, Z. H.; Wang, D. L.; Wang, W. U.; Lieber, C. M. Nano Lett. 2003, 3, 149. (7) Huang, Y.; Duan, X. F.; Cui, Y.; Lauhon, L. J.; Kim, K. H.; Lieber, C. M. Science 2001, 294, 1313. (8) He, J.; Durrani, Z. A. K.; Ahmed, H. Microelectron. Eng. 2004, 73, 712. (9) Cui, Y.; Wei, Q. Q.; Park, H. K.; Lieber, C. M. Science 2001, 293, 1289. (10) Wagner, R. S.; Ellis, W. C. Appl. Phys. Lett. 1964, 4, 89. (11) Cui, Y.; Lauhon, L. J.; Gudiksen, M. S.; Wang, J.; Lieber, C. M. Appl. Phys. Lett. 2001, 78, 2214. (12) Wu, Y.; Cui, Y.; Huynh, L.; Barrelet, C. J.; Bell, D. C.; Lieber, C. M. Nano Lett. 2004, 4, 433.
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