Self-Assembly of Ordered Epitaxial Nanostructures on Polygonal

Jan 16, 2013 - epitaxially grown on a polygonal cross-section nanowire (PCS-. NW) using both theoretical and phase-field modeling. Our studies show th...
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Self-Assembly of Ordered Epitaxial Nanostructures on Polygonal Nanowires Liang-Xing Lu, M. S. Bharathi, and Yong-Wei Zhang* Institute of High Performance Computing, A*STAR, Singapore 138632 S Supporting Information *

ABSTRACT: We study the self-assembly of ordered nanostructures, that is, nanorings (NRs) and quantum dots (QDs), epitaxially grown on a polygonal cross-section nanowire (PCSNW) using both theoretical and phase-field modeling. Our studies show that, by increasing the PCS-NW size, transitions from ordered NRs to ordered QDs on facets and further to ordered QDs on ridges occur. The predicted morphologies and their transitions are in excellent agreement with existing experiments. Our study suggests a novel approach to fabricate ordered nanostructures on nanowires.

KEYWORDS: Core−shell nanowire, nanoring, quantum dot, phase field simulation, size effect

S

Guo et al.25 These modeling works suggest that geometric parameters, such as core radius and shell thickness, play a crucial role in determining the stability of the shell surface. Though polygonal cross-section nanowires (PCS-NWs) have been observed experimentally in several material systems, an indepth understanding of their self-assembly, ordering, and stability is still lacking, and no theoretical analysis and numerical modeling on PCS-NWs have been reported. To control and design these hybrid nanowire-based structures and devices, an in-depth understanding of the formation, selfassembly, and stability conditions for these nanostructures is necessary and is the subject of this work. In this paper, we first develop theoretical analysis to identify different ordered patterns of nanostructures on a PCS-NW with different core sizes. We then perform phase field simulations to enrich our understanding beyond our theoretical analysis. Our objectives are: (1) to understand the various ordered morphologies exhibited by heteroepitaxial core−shell PCSNWs; and (2) to identify the system parameters that control these ordered morphologies. Theoretical Analyses. Two-Dimensional (2D) Analysis. Experimental results20−22 suggest that surface nanostructures with different morphologies may occur when epitaxially grown on PCS-NWs. We hypothesize that there exist transitions between different morphologies, and these transitions arise from bifurcations in the free energy profile as a function of the core size. For our energetic analysis in 2D, as an example, we

emiconductor nanowires (NWs) hold great promise for high-performance functional nanoelectronic devices,1,2 sensors,3,4 thermoelectric devices,5,6 high-mobility field effect transistors,7 batteries,8,9 photodetectors and solar cells,10,11 owing to their fascinating properties in electronics, photonics, magnetics, and so on. Currently, an important research topic for NWs is to synthesize hybrid nanostructures via integrating different materials on a single NW. Such integrations allow for formation of artificial nanostructures on a single NW with fascinating multifunctional properties that cannot be found in nature.12,13 Epitaxial deposition of heteromaterials with a lattice mismatch has been a natural way to synthesize hybrid nanostructures.14−17 The lattice mismatch between the NW core and the shell deposited on its surface provides an energetic driving force for the shell surface to spontaneously selfassemble into various nanostructures via the Stranski− Krastanov (S−K) mechanism.18−20 On cylindrical NWs, selfassembly of Ge QDs via the S−K mechanism was observed by Pan et al.18 and Goldthorpe et al.19 On faceted NWs, the growth of nanorings (NRs) was reported by Paladugu et al.20 and by Uccelli et al.,21 and growth of QDs was reported by Uccelli et al.21 and Yatago et al.22 In ref 21, it was observed that the formation of InAs QDs on both facets and ridges of the hexagonal GaAs NW, and the formed QDs were aligned in a chain-like fashion along the facets or ridges of the NW. In ref 22, MnAs QDs were observed to grow at ridges for the 100 nm diameter GaAs NW while on both ridges and facets for the 300 nm diameter GaAs NW. Morphological evolution of nanostructures on cylindrical NWs was studied using perturbation analysis by Schmidt et al.23 and Wang et al.24 and using finite element dynamic analysis by © 2013 American Chemical Society

Received: November 3, 2012 Revised: December 29, 2012 Published: January 16, 2013 538

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where a is the side length of the square core, and S and V are nondimensional coefficients, which are functions of only geometry. The values of S and V are expressed as S1, S2, S3 and V1, V2, V3 for Confs 1, 2, and 3, respectively. It is obvious that the surface area of Conf 1 is smaller than that of Conf 2 and the surface area of Conf 2 is smaller than that of Conf 3. So we have the following relationship: S1 < S2 < S3. For the specific geometries given in Supporting Information (see Figure S1), we have S1 = 2.36, S2 = 2.51, and S3 = 4.29. Since a QW located at a corner can relax more strain than a QW located on a sidesurface, then we have V1 > V2 > V3. For specific geometries given in Supporting Information (see Figure S1), we have V1 = 0.32, V2 = 0.23, and V3 = 0.10. By combining eg and eel together, we can plot the total energy density curves for each configuration as a function of side length. In Figure 1b, dashed lines in blue, red, and purple are energy density curves corresponding to Confs 1, 2, and 3, respectively; while the solid line is the global minimum energy density curve. It can be seen from Figure 1b that the three energy curves are intersecting at two core sizes, that is, acri1 and acri2, which divide the core size into three regions. For a < acri1, Conf 1 is the most energetically favorable one; for acri1 < a < acri2, Conf 2 is the most energetically favorable one; while for a > acri2, Conf 3 is the most energetically favorable one. The values of acri1 and acri2 can be obtained from eqs 1 and 2 as:

consider a PCS-NW with a square core and the three possible shell configurations as shown in Figure 1. In Configuration 1

Figure 1. 2D analysis for the transitions of different shell configurations. (a) Schematic of three shell configurations. (b) Energy density curves of the three configurations as a function of core size a. Dashed lines of blue, red, and purple correspond to Conf 1, Conf 2, and Conf 3 in a, respectively, and the solid line is the minimum energy curve.

(1)

eel = V(geometry) ·με02

(2)

(3)

⎛ S − S2 ⎞ γ acri2 = ⎜ 3 ⎟· ⎝ V2 − V3 ⎠ με02

(4)

The exact values of these critical sizes are dependent upon the precise geometry parameters. For example, using the geometries as shown in Figure 1S, and taking γ = 1.23 J/m2, μ = 42 GPa,26 and ε0 = 4%, acri1 and acri2 are found to be about 29 and 252 nm, respectively. Clearly, these values are in the same order of magnitude with experimental results.20−22 Similar calculations were also performed using other contact angles. The calculations indicate that the change in contact angle only leads to a change in the values of acri1 and acri2, but the qualitative energy profile as shown in Figure.1b remains unchanged. In general, the critical transition size can be written as, γ acri = K(geometry change)· 2 με0 (5)

(Conf 1 in Figure 1), the shell and core together form a perfect cylinder as shown in Figure 1a. In Configuration 2 (Conf 2), the shell breaks into periodical quantum wires (QWs) along the axial direction with each QW sitting one of four side-surfaces of the core as shown in Figure 1b. While in Configuration 3 (Conf 3), the shell breaks into QWs along the axial direction with each QW sitting on one of the four corners of the core as shown in Figure 1c. NWs with other polygonal cross sections can be analyzed in the same manner. In the following, we assume that all of these three configurations have the same shell volume and core size. The total energy of each configuration includes interface energy between shell and core, surface energies of shell and core, and elastic energy induced from the mismatch strain between shell and core. For the sake of simplicity, we ignore the interface energy between shell and core and assume that the materials of core and shell have the same surface energy and elastic properties. We denote the surface energy density as γ, the shear modulus as μ, and the mismatch strain as ε0. Since surface energy is proportional to surface area, and elastic energy proportional to volume, the total surface energy (eg) and elastic energy (eel) per unit volume along the NW should have the following expressions: eg = S(geometry) ·γ /a

⎛ S − S1 ⎞ γ acri1 = ⎜ 2 ⎟· ⎝ V1 − V2 ⎠ με02

which states that the critical transition size is proportional to the surface energy density and inversely proportional to the strain energy density με20, with a proportionality coefficient K dependent only on the geometrical change. Three-Dimensional (3D) Analysis. In our 3D analysis, we also consider three possible shell structures on a NW core with a square cross-section shown in Figure 2, in direct analogy to our 2D analysis. In Configuration a (Conf a in Figure 2), the shell forms periodical nanorings (NRs) as shown in Figure 2a. In Configuration b (Conf b), the shell breaks into periodical QDs. Within each period, four QDs are formed with each sitting on one of the four facets of the nanowire. In Configuration c (Conf c), four QDs are formed with each sitting on one of the four ridges. As in our 2D model, the morphological evolution of the shell in 3D is also dominated by the competition between surface energy and elastic energy and can be controlled by changing the core size. For a given NW 539

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Phase Field Modeling. Our energy calculations clearly point out the existence of transitions between different ordered configurations with the core size. In the following, we use a phase field model (see Supporting Information for the detailed description of the model) to study the kinetic pathway and stability of quantum dots formed during the morphological evolution of core−shell NWs via the S−K mechanism. The Ge/ Si core−shell system is used as a model system. For this system, the lattice mismatch strain is about 4%, the interface energies are taken to be γ12 = 1.23 N/m, γ13 = 1.23 N/m, and γ23 = 0 N/ m,26 where subscript 1 denotes the vapor phase, 2 denotes the Ge phase, and 3 denotes the Si phase. We assume the elastic properties of both Ge and Si are isotropic and have the same elastic constants, with shear modules μ = 42.1 GPa and Poisson’s ratio ν = 0.27. 2D Simulation. To compare with the 2D energetic analysis results, we study the morphological evolution of shell epitaxially deposited on a core with a square cross section. Simulation area is discretized into 128 × 128 regular grids with each grid length being about 9 nm in real space. Before the deposition, a thin layer of 18 nm (2 grids) thickness was added on the core to model the wetting effect. A random white noise with 9 nm (1 grid) amplitude was imposed on the thin layer to add in randomness to the growth. The deposition rate is set as 0.8 nm per unit simulation time. This deposition rate was chosen to allow sufficient diffusion of shell material to reach relatively lower energy positions. Figure 3a−c shows the snapshots of morphological evolution of shell material on a NW with a = 45, 189, and 279 nm,

Figure 2. 3D analysis for the transitions of shell configurations. (a) Schematics of the three configurations, Conf a corresponding to Conf 1 in 2D, Conf b corresponding to Conf 2 in 2D, and Conf c corresponding to Conf 3 in 2D. (b) Energy density curves of the three configurations as a function of core size a. Dashed lines of blue, red, and purple correspond to Conf a, Conf b, and Conf c, respectively, and the solid line is the minimum energy curve.

with side length a, the surface energy and elastic energy per volume have the same expressions as the 2D cases given by eqs 1 and 2, respectively. Values of S and V corresponding to Confs a, b, and c are denoted as Sa, Sb, Sc and Va, Vb, Vc, respectively. Same as our 2D cases, we also have the following relationships: Sa < Sb < Sc and Va > Vb > Vc. For the geometries given in Figure S2 in the Supporting Information, we have Sa = 3.14, Sb = 3.63, Sc = 3.91 and Va = 0.30, Vb = 0.19, Vc = 0.14. In our calculations, γ = 1.23 J/m2, μ = 42 GPa,26 and ε0 = 4%. Total energy density curves for these three 3D configurations as a function of a are plotted in Figure 2b. As in the 2D analyses, we see two transition sizes: acri1 and acri2. For a < acri1, Conf a is the most energetically preferred one; Conf b is the most energetically preferred one for acri1 < a < acri2, and for a > acri2, Conf c is the most energetically preferred one. In the above cases, the real values for acri1 and acri2 are 82 and 103 nm, respectively, which are also in the same order of magnitude with existing experimental results.20−22 It should be noted that QDs in the 3D configurations can be arranged in different ways on the PCS-NW surfaces. For example, the four QDs arranged in a staggered manner as shown in Conf b in Figure 2a can also be arranged in a coplanar manner on the same cross section plane. Similarly, the four QDs arranged in a staggered manner as shown in Conf c in Figure 2a can also be arranged in a coplanar manner on the same cross section plane. Our finite element analyses show that compared to the staggered arrangement, the coplanar arrangement always has a higher energy. This is why, in the present work, we only focus on the staggered arrangement of QDs.

Figure 3. 2D phase field simulations of shell grown on a square core with different side lengths. (a) 45 nm, (b) 189 nm, and (c) 279 nm.

respectively, at t = 50. For a = 45 nm as shown in Figure 3a, shell material grows around the core and forms a cylinder, similar to Conf 1 shown in Figure 1. For a = 189 nm as shown in Figure 3b, shell material aggregates on the side surface and eventually forms four QWs with each on one side-surface, similar to Conf 2 in Figure 1. For a = 279 nm, shell material gathers at the corners of the core, and eventually forms four QWs with each sitting at one of the corners, similar to Conf 3 in Figure 1. The 2D phase field simulations give critical transition sizes of acri1 and acri2 about 10 grids and 29 grids, which are about 90 and 261 nm in real space, respectively. These values are in the same order of magnitude with the ones obtained from the 2D model given in Figure 1. We can further increase the core size beyond acri2 to examine the self-assembly of shell material. Our phase field simulation suggests that there exists another transition point at about 36 grids, i.e., at about 324 nm in real space, beyond which QWs form both on the facets and corners. A typical result is given in Figure S3 in Supporting Information. This result is in good agreement with the observation by Yatago et al.22 540

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the same ridge results in the formation of bigger QDs on the ridge (see t = 50 in panel c of Figure 4). This configuration is the same as Conf c shown in Figure 2. Our 3D phase field simulations show that the morphological transition from Conf a to Conf b occurs at a = 9 grids, which is about 81 nm in real space, and at a = 12 grids for transition from Conf b to Conf c, which is about 108 nm in real space. These critical values are in excellent agreement with that from the 3D analysis given in Figure 2. Same as 2D simulation, our 3D phase field simulations show that there also exists another transitional point at about 180 nm (20 grids). At this point, shell material will experience a further transition from Conf c to a mixed configuration of Conf b and Conf c; that is, QDs appear both on facets and ridges as shown in Figure S6a and b in the Supporting Information. We also studied the morphological evolution of shell deposited on hexagonal cross section NWs. Similar morphologies and their transitions were also observed. Figures 5a−d

Besides the morphological evolution of shell under growth condition, we also examined the morphological evolution under annealing condition. Simulations under annealing condition gives the similar tendency as that under growth condition: With an increase in core size, the equilibrium morphology of shell also experiences transitions from Conf 1 to Conf 2 and then to Conf 3 as shown in Figures 1 and 3. Besides square cross-section cores, we also carried out simulations on hexagonal and octagonal cross section cores. The same transition patterns were also observed. Typical results are shown in Figure S4 in Supporting Information. The same transition patterns obtained from different polygonal core shapes imply that the transitions shown in Figure 1 may be a common feature in the growth of shell on a polygonal core. 3D Simulation. We simulated the morphological evolution during the growth of shell material deposited on a square core with a = 45, 99, and 117 nm. Figure 4a−c shows the snapshots

Figure 4. 3D phase field simulations of shell grown on the surface of square cross section nanowire with different core sizes. (a) 45 nm, (b) 99 nm, and (c) 117 nm. Color represents the shell thickness.

Figure 5. 3D phase field simulations of shell grown on the surface of a hexagonal cross section nanowire with different core sizes. (a) 63 nm; (b and c) 135 nm; and (d) 171 nm. Color represents to shell thickness.

at t = 50 (see the time series in Figure S5 in Supporting Information). Similar with our 2D simulations, we apply an initial shell layer with 18 nm thickness to consider the wetting effect and 9 nm random fluctuations to add randomness to the growth process. Color in Figure 4 represents the distance from shell surface to the core−shell interface. The region in blue represents the core nanowire. For deposition on NW with side length of 45 nm as shown in Figure 4a, the shell breaks up and self-assembles into ordered NRs along the axial direction. This result is in excellent agreement with the 3D analysis presented in Figure 2a. For deposition on NW with side length of 99 nm, the shell breaks up and self-assembles into ordered QDs on facets. This QD pattern is in good agreement with the 3D analysis given in Figure 2 although the shape of QDs obtained is slightly different from the spherical shape used in the 3D energetic analysis. This nonspherical shape may come from the different length scales in the x−y plane and z direction. Due to the smaller dimension in x-axis and y-axis, stress can be relaxed more easily in the x−y plane than along the z-axis, causing the elongated island shape. For deposition of shell material on NW with side length of 117 nm, shell material breaks into two columns of QDs on each facet at the early stage (see t = 10 in Figure S5). These two columns of QDs align themselves parallel to the ridges in a staggered manner on the side surface, similar to the observations made by Uccelli et al.21 in the growth of InAs on GaAs NWs. Clearly, the double-column configuration corresponds to the growth of a large core size. As the deposition continues, the QDs formed at the early stage grow bigger and start to migrate toward the ridges of NW as shown in t = 30 of Figure S5. The merging of QDs adjacent to

show a typical case. Panel a shows the snapshot at t = 50 of shell deposited on a hexagonal NW with a diameter of 63 nm. It can be seen that ordered NRs are formed along the axial direction of the NW. Panel b shows the case with NW diameter of 135 nm. It is seen that the shell breaks up into perfectly ordered QDs located on the facets. Panel c also shows the case with the same NW, but with a slightly different initial surface perturbation. It is seen that instead of forming perfectly ordered QDs, some defects are present. Hence, the ordering of QDs is also influenced by surface and growth conditions. Overall the QD array is still close to the perfect one as shown in Panel b. We further perform annealing simulation to the configuration shown in Panel c, and no obvious change of QD morphology is observed, indicating the configuration c is in metastable state (see Section S.6 of the Supporting Information for details). Panel d shows the case with core diameter of 171 nm. In this case, QDs are located at the ridges. The two-column configuration on the same facets in the early deposition stage is also observed (around t = 10). Eventually, the QDs on ridges self-assemble themselves into a relatively ordered pattern. Although the QDs on the ridges are similar to that predicted in our energy analysis, they are not in a perfect array. Upon further annealing, we observed the small extra dot merges with a larger one next to it and forms a more ordered structure (see Section S.6 of the Supporting Information for details). Similar with the square NW, a further increase in the side length to 279 nm results in the formation of QDs on both facets and ridges as shown in Figure S6b. Our phase field simulations reveal that 541

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(2) Duan, X. F.; Huang, Y.; Cui, Y.; Wang, J. F.; Lieber, C. M. Nature 2001, 409, 66. (3) Wang, W. U.; Chen, C.; Lin, K. H.; Fang, Y.; Lieber, C. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 3208. (4) Zheng, G. F.; Patolsky, F.; Cui, Y.; Wang, W. U.; Lieber, C. M. Nat. Biotechnol. 2005, 23, 1294. (5) Hochbaum, A. I.; Chen, R. K.; Delgado, R. D.; Liang, W. J.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. D. Nature 2008, 451, 163. (6) Boukai, A. I.; Bunimovich, Y.; Tahir-Kheli, J.; Yu, J. K.; Goddard, W. A.; Heath, J. R. Nature 2008, 451, 168. (7) Cui, Y.; Zhong, Z. H.; Wang, D. L.; Wang, W. U.; Lieber, C. M. Nano Lett. 2003, 3, 149. (8) Chan, C. K.; Peng, H.; Liu, G.; McIlwrath, K.; Zhang, X. F.; Huggins, R. A.; Cui, Y. Nat. Nanotechnol. 2008, 3, 31. (9) Chan, C. K.; Zhang, X. F.; Cui, Y. Nano Lett. 2008, 8, 307. (10) Kayes, B. M.; Atwater, H. A.; Lewis, N. S. J. Appl. Phys. 2005, 97, 114302. (11) Tian, B. Z.; Zheng, X. L.; Kempa, T. J.; Fang, Y.; Yu, N. F.; Yu, G. H.; Huang, J. L.; Lieber, C. M. Nature 2007, 449, 885. (12) Shevchenko, E. V.; Talapin, D. V.; Kotov, N. A.; O’Brien, S.; Murray, C. B. Nature 2006, 439, 55. (13) Lee, J.; Hernandez, P.; Govorov, A. O.; Kotov, N. A. Nat. Mater. 2007, 6, 291. (14) Petroff, P. M.; DenBaars, S. P. Superlattices Microstruct. 1994, 15, 15. (15) Borgström, M. T.; Zwiller, V.; Müller, E.; Imamoglu, A. Nano Lett. 2005, 5, 1439. (16) Minot, E. D.; Kelkensberg, F.; Kouwen, M.; Dam, J. A.; Kouwenhoven, L. P.; Zwiller, V.; Borgström, M. T.; Wunnicke, O.; Verheijen, A. V.; Bakkers, E. P. A. M. Nano Lett. 2007, 7, 367. (17) Tribu, A.; Sallen, G.; Aichele, T.; André; Poizat, J.; Bougerol, C.; Tatarenko, S.; Kheng, K. Nano Lett. 2008, 8, 4326. (18) Pan, L.; Lew, K. K; Redwing, J. M.; Dickey, E. C. Nano Lett. 2005, 5, 1081. (19) Goldthorpe, I. A.; Marshall, A. F.; Mclntyre, P. C. Nano Lett. 2008, 8, 4081. (20) Paladugu, M.; Zou, J.; Guo, Y. N.; Zhang, X.; Hannah, J. J.; Gao, Q.; Tan, H. H.; Jagadish, C.; Kim, Y. Angew. Chem., Int. Ed. 2009, 48, 780. (21) Uccelli, E.; Arbiol, J.; Morante, J. R.; Morral, A. F. I. ACS Nano 2010, 4, 5985. (22) Yatago, M.; Iguchi, H.; Sakita, S.; Hara, S. Jpn. J. Appl. Phys. 2012, 51, 02BH01. (23) Schmidt, V.; Mclntyre, P. C.; Gösele, U. Phys. Rev. B 2008, 77, 235302. (24) Wang, H.; Upmanyu, M.; Ciobanu, C. V. Nano Lett. 2008, 8, 4305. (25) Guo, J. Y.; Zhang, Y. W.; Vivek, B. S. ACS Nano 2010, 4, 4455. (26) Jesson, D. E.; Pennycook, S. J.; Baribeau, J. M.; Houghton, D. C. Phys. Rev. Lett. 1993, 71, 1744. (27) Seol, D. J.; Hu, S. Y.; Liu, Z. K.; Chen, L. Q.; Kim, S. G.; Oh, K. H. J. Appl. Phys. 2005, 98, 044910. (28) Wang, Y. U.; Jin, Y. M.; Khachaturyan, A. G. J. Appl. Phys. 2002, 91, 6435. (29) Eshelby, J. D. Proc. R. Soc. London, Ser. A 1957, 241, 376. (30) Wang, Y. U.; Yongmei, M. J.; Khachaturyan, A. G. Acta Mater. 2004, 52, 81.

transitions of hexagonal cross section NWs take place at a = 81 nm (9 grids) from NRs to QDs on facets, and at a = 171 nm (19 grids) from QDs on facets to QDs on ridges, and at a = 270 nm (30 grids) from QDs on ridges to the formation of mixed morphology. These results are in excellent agreement with the experimental observation of Yatago et al.28 for MnAs grown on hexagonal GaAs NWs. Self-assembly of QDs on a PCS-NW is a complex process governed by both growth kinetics and thermodynamics. Although our energy analysis indicates the presence of ordered QD arrays and our field phase simulations also show the formation of the ordered arrays of QDs, often defected QD arrays are formed. Our explanation for the formation of defects is that the thermodynamic driving force for the QD selfassembly is relatively weak, and hence kinetic factors, such as growth conditions and initial surface conditions, also play a role in the QD self-assembly and ordering. This is similar to the presence of vacancy and interstitial in a crystalline material. Importantly, our phase field simulations (see Section S.6 of the Supporting Information for details) suggest that, upon annealing, these defected QD arrays are able to evolve toward more regular and ordered state, rather than undergo coarsening. Conclusions. We first performed theoretical analyses in 2D and 3D to study the transitions of ordered nanostructure morphologies heteroepitaxially grown on the surface of polygonal cross-section NWs. The analyses predict two morphological transitions with increasing the core size in both 2D and 3D. If the core size is smaller than a critical size, shell tends to form a cylinder in 2D case and NRs in 3D case. When the core size exceeds the critical size, QWs in 2D and QDs in 3D will appear. The QWs and QDs will sit on the facets if the core size is smaller than another critical size, and sit on corners or ridges if the core size is larger than the critical size. Besides the theoretical analyses, we also developed a phase field model to simulate the growth/annealing process on square and hexagonal NWs. The phase field simulation results fully support the energetic analyses and are in good agreement with existing experimental observations. It is noted that both thermodynamic and kinetic factors play an important role in the self-assembly and ordering of QDs on PCS-NWs. As a consequence, defected QD arrays are often formed. Our phase field simulations show that annealing can improve the ordering of QDs and lead to a more regular QD array. Our present work suggests a novel approach to grow stable and ordered nanostructures epitaxially on polygonal nanowires.

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ASSOCIATED CONTENT

S Supporting Information *

Phase field model, geometry parameters for FE analyses, and more figures of phase field simulation. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +65 64191478; fax: +65 64674350. E-mail: zhangyw@ ihpc.a-star.edu.sg (Y.W. Zhang). Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Cui, Y.; Lieber, C. M. Science 2001, 291, 851. 542

dx.doi.org/10.1021/nl3040543 | Nano Lett. 2013, 13, 538−542