REPORTS ous approaches can now be described as 13. O. Albert, C.A. Gautier, J. C Loulergue, ]. Etchepare, 229(1973). Data for n in the lows In Fig. 4, A and C, the angle between t available at temperatures above 90 not its magnitude. In the absence of dispersion, 15. G. P. wiederrecht et al., Phys. Rev. B 51, 916(1995). and the value of n at the gap[B this leads to a shock-wave singularity because 16. H P. Perry, T..DoughteryPhysRev.8.5778 frared Phys. 23, 359(1983). The latter was used an adjustable parameter to fit the time-domain data of the constructive interference of waves of 17. R.M. Koehl, s. Adachi, K. A Nelson, J. Chem. Phys. 24. T. E. Stevens, thesis, University of Michigan,Ann arbitrary g In Fig 4B, both the angle and q are predetermined because the components of q 18. The calculations were performed with MATH- 25. M. Born, K Huang Dynamical Theory of Crystal Lat- orthogonal to e, are set by the grating. These MATICA Version 4.0(Wolfram Research, Inc, Cham- 26. We used heterodyne detection methods. The pump considerations apply to the imaging experi- hes differ in that e=o outside the a ments of(17)and to early work on polariton for the superluminal case, whereas, for v co, the propagation(27) where the direction of motion tensity does not vanish notwithstanding a substan- with a polarizing beam cube and spect can be identified with that of the Cherenkov (24). The probe beam and a refere ispersed by the spectrometer, were focu ression. Finally, consider single-pump exci tation when the source lateral dimensions are ane)at v co contradicts the phase-velocity argu- ured with a dual channel EG&G 5 sufficiently large that the wave vectors of the er to get△T nent predicting ec= 0 because c/nv can be arDl- 27. G M. Gale, F Vallee, C. Flytzanis, Phys. Rev. Lett. 57. pump pulse and the polariton are nearly col linear. In this case, the phase-matching condi- no). the Cherenkov angle for v co goes all the wi /2 atv= 0. Also note that de/dn oc at eling grating were interpreted in terms of Because q s n2/co at low frequencies, it is clear hich, according to Eq. 4. avoided mode crossings and anharmonic 0. Hence, at subluminal speeds, radiation at these that Cherenkov field profiles can be fairly that phase matching can only be attained at ed, there may be a need to compare the expe subluminal speeds, in agreement 6: see 21. See, e.g. Y.R.Shen, The Principles of Nonlin Fig. 2C. Under the same(quasi-planar) condi- (Wiley, New York, 1984). Materials with 29. V. P. Zrelov, J. Ruzicka, A. A. Tyapkin, JuNR Rapid ons, and not too far from the pump pulse, the field for the grating method results from the density become interference between two polaritons at, say,qx E+Rim)Q). where Eg and Mare electric C. Aku-Leh for assistance with the = +2m/t and q =0, where t is the grating mponents of the pump pulse and that of the alculations and to D G. Steel for critical reading of spacing(12-17). This and Eq. 7 give qt the manuscript. T.E.S. would like to thank the Planck-Institut fur Festkorpertorschung for warm sing (2- vo)). This field is represented in Fig. 22. a ui. 4. Phys. Chem. Ref Data 13, 102(1984) bR928yp四ym 4B by the rectangle with the checkerboard pat asearch under contract F49620-00-1-0328 ten(28) through the Multidisciplinary University Re (22)and at 2 K A GobeL, T Ruf, ]. M. Zhang, RLau tiative program. M Cardona, Phys. Rev. B 59, 2749(1999): T. Hattori, Y. Homma, A. Mitsuishi, M. Tacke, Opt. Commun. 7. 29 September 2000: accepted 30 November 2000 1. v. P. Zrelow Radiation in High-Energy 2. T. Ypsilantis, ]. Seguinot, Nucl. Instrum. Methods A Directed Assembly of 433 3. P. A. Cherenkov. DokL. Akad. Nauk SSSR 2. 451 (1934) One-Dimensional Nanostructures 4. S Vavilov, Dokl. Akad. Nauk SSSR 2, 457(1934). 5. L Frank, Vavilov-Cherenkov Radiation(Nauka, Mos into Functional Networks cow,1988 L Tamm, Dokl Akad. Nauk SSSR 14, 109 Yu Huang, *Xiangfeng Duan, Qingqiao Wei, 7. Expressions for the Cherenkov field in dispersive m dia were first derived by L. Tamm [ Phys. 1. 42 Charles M. Lieber,2+ (1939)and independently by E Fermi(Phys. Rev. 57. 485(1940), who used the Lorentz(single-pole)ap- One-dimensional nanostructures, such as nanowires and nanotubes, represent reduction in the energy loss, a phenomenon kr the smallest dimension for efficient transport of electrons and excitons and thus the density effect. Ignoring dissipation, the are ideal building blocks for hierarchical assembly of functional nanoscale tion for Chere electronic and photonic structures. We report an approach for the hierarchical nance, a partide can in principle radiate at ssembly of one-dimensional nanostructures into well-defined functional net works. We show that nanowires can be assembled into parallel arrays with G. N. Afanasiev, V. G. Kartavenko, E. N. Magar control of the average separation and, by combining fluidic alignment with surface-patterning techniques, that it is also possible to control periodicity In ork was motivated in part by recent CI addition, complex crossed nanowire arrays can be prepared with layer-by-layer 9. To the best of our knowledge, the nume assembly with different flow directions for sequential steps. Transport studies st to uncover diffe show that the crossed nanowire arrays form electrically conducting networks, with individually addressable device function at each cross point. surfaces at which the field exhibits may 10. D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A Nanoscale materials, for example, nanoclus- rication (1-4). Research focused on zero- Phys. Rev. Leti 65(1984), and ref ters and nanowires(NWs), represent attrac- dimensional nanoclusters has led to substan- tive building blocks for hierarchical assembly tial advances, including the assembly of ar 11. D. A Kleinman, D H. Auston, IEEE J. tron.QE20,964(1984 of functional nanoscale devices that could rays with order extending from nanometer to 12. P. Grenier, D Houde, S JandL, L A Boatner, Phys overcome fundamental and economic limita- micrometer length scales(4-9). In contrast, B50.16295(1994) tions of conventional lithography-based fab- the assembly of one-dimensional (ID)nano- 630 26JanUary2001Vol291ScieNcewww.sciencemag.orgous approaches can now be described as fol￾lows. In Fig. 4, A and C, the angle between the polariton wave vector and the z axis is fixed, but not its magnitude. In the absence of dispersion, this leads to a shock-wave singularity because of the constructive interference of waves of arbitrary q. In Fig. 4B, both the angle and q are predetermined because the components of q orthogonal to ez are set by the grating. These considerations apply to the imaging experi￾ments of (17) and to early work on polariton propagation (27) where the direction of motion can be identified with that of the Cherenkov expression. Finally, consider single-pump exci￾tation when the source lateral dimensions are sufficiently large that the wave vectors of the pump pulse and the polariton are nearly col￾linear. In this case, the phase-matching condi￾tion is q ' V[n(vL) 1 vLn˙(vL)]/c [ V/cg (vL). Because q ' V/c0 at low frequencies, it is clear that phase matching can only be attained at subluminal speeds, in agreement with Eq. 6; see Fig. 2C. Under the same (quasi-planar) condi￾tions, and not too far from the pump pulse, the field for the grating method results from the interference between two polaritons at, say, qx 5 62p/, and qy 5 0, where , is the grating spacing (12–17). This and Eq. 7 give qz , 5 2p(n2 v2 /c2 2 1)21/2, leading to E ; sin(2px/,) sin[qz (z 2 vt)]. This field is represented in Fig. 4B by the rectangle with the checkerboard pat￾tern (28). References and Notes 1. V. P. Zrelov, Cherenkov Radiation in High-Energy Physics (Israel Program for Scientific Translations, Jerusalem, 1970). 2. T. Ypsilantis, J. Seguinot, Nucl. Instrum. Methods A 433, 1 (1999), and references therein. 3. P. A. Cherenkov, Dokl. Akad. Nauk SSSR 2, 451 (1934). 4. S. Vavilov, Dokl. Akad. Nauk SSSR 2, 457 (1934). 5. I. Frank, Vavilov-Cherenkov Radiation (Nauka, Mos￾cow, 1988). 6. iiii, I. Tamm, Dokl. Akad. Nauk SSSR 14, 109 (1937). 7. Expressions for the Cherenkov field in dispersive me￾dia were first derived by I. Tamm [ J. Phys. 1, 439 (1939)] and independently by E. Fermi [Phys. Rev. 57, 485 (1940)], who used the Lorentz (single-pole) ap￾proximation to show that polarization effects lead to a reduction in the energy loss, a phenomenon known as the density effect. Ignoring dissipation, the condi￾tion for Cherenkov emission reads v 2 . c 2/«(V). Because «(V) can reach large values in the proximity of a resonance, a particle can in principle radiate at arbitrarily small velocities. 8. G. N. Afanasiev, V. G. Kartavenko, E. N. Magar, Physica B 269, 95 (1999), and references therein. This work was motivated in part by recent CR experiments with CERN’s high-energy beam of lead ions (29, 30). 9. To the best of our knowledge, the numerical study of Afanasiev et al. (8) was the first to uncover differences between superluminal and subluminal CR. In their work, these differences manifest themselves in the separate open (v . c0) and closed (v , c0) behavior of the surfaces at which the field exhibits maxima. 10. D. H. Auston, K. P. Cheung, J. A. Valdmanis, D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984), and ref￾erences therein. 11. D. A. Kleinman, D. H. Auston, IEEE J. Quantum Elec￾tron. QE-20, 964 (1984). 12. P. Grenier, D. Houde, S. Jandl, L. A. Boatner, Phys. Rev. B 50, 16295 (1994). 13. O. Albert, C. A. Gautier, J. C. Loulergue, J. Etchepare, Solid State Commun. 107, 567 (1998). 14. H. J. Bakker, S. Hunsche, H. Kurz, Rev. Mod. Phys. 70, 523 (1998), and references therein. 15. G. P. Wiederrecht et al., Phys. Rev. B 51, 916 (1995). 16. H. P. Perry, T. P. Doughtery, Phys. Rev. B 55, 5778 (1997). 17. R. M. Koehl, S. Adachi, K. A. Nelson, J. Chem. Phys. 110, 1317 (1999). 18. The calculations were performed with MATH￾EMATICA Version 4.0 (Wolfram Research, Inc., Cham￾paign, IL, 1999). 19. The two regimes differ in that E § 0 outside the cone for the superluminal case, whereas, for v , c0, the intensity does not vanish notwithstanding a substan￾tial drop at the boundary. Hence, there is a shock￾wave singularity at superluminal but not at sublumi￾nal speeds. The presence of a cone (as opposed to a plane) at v , c0 contradicts the phase-velocity argu￾ment predicting uC 5 0 because c/nv can be arbi￾trarily close to unity. 20. Unlike the superluminal case, in which uC # cos21(1/ n0), the Cherenkov angle for v , c0 goes all the way to uC 5 p/2 at v 5 0. Also note that d«/dV 3 ` at V5VC, V0, which, according to Eq. 4, leads to a 5 0. Hence, at subluminal speeds, radiation at these frequencies is concentrated at r 5 0. 21. See, e.g., Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984). Materials with infrared￾active vibrations, such as ZnSe, show an additional contribution to the nonlinear polarization due to the combined phonon-photon character of the eigen￾modes. The energy density becomes 2 S%i %k (xikm (2) Em 1 Rik (m) Qm), where %K and Em are electric field components of the pump pulse and that of the infrared radiation, Qm is the amplitude of a phonon component, and R(m) is its associated Raman tensor. 22. H. Li, J. Phys. Chem. Ref. Data 13, 102 (1984). 23. Values of the infrared constants of ZnSe from the literature exhibit considerable scattering. Those in Fig. 2 were interpolated from measurements at 80 K (22) and at 2 K [A. Go¨bel, T. Ruf, J. M. Zhang, R. Lauck, M. Cardona, Phys. Rev. B 59, 2749 (1999); T. Hattori, Y. Homma, A. Mitsuishi, M. Tacke, Opt. Commun. 7, 229 (1973)]. Data for n in the visible range are available at temperatures above 90 K (22). Values at 10 K were obtained from an expression giving n(V) in terms of the band gap, the carrier effective masses, and the value of n at the gap [B. Jensen, A. Torabi, Infrared Phys. 23, 359 (1983)]. The latter was used as an adjustable parameter to fit the time-domain data. 24. T. E. Stevens, thesis, University of Michigan, Ann Arbor (2000). 25. M. Born, K. Huang, Dynamical Theory of Crystal Lat￾tices (Clarendon, Oxford, 1996), p. 100. 26. We used heterodyne detection methods. The pump beam was modulated with a mechanical chopper at 3.51 kHz. The transmitted probe light was analyzed with a polarizing beam cube and spectrally resolved (24). The probe beam and a reference beam, also dispersed by the spectrometer, were focused onto separate photodiodes, and their voltage difference was measured with a dual channel EG&G 5320 digital lock-in amplifier to get DT. 27. G. M. Gale, F. Valle´e, C. Flytzanis, Phys. Rev. Lett. 57, 1867 (1986). 28. In (14), pump-probe data for polaritons generated with a traveling grating were interpreted in terms of avoided mode crossings and anharmonicity. Given that Cherenkov field profiles can be fairly complicat￾ed, there may be a need to compare the experiments with calculations of the field due to a finite traveling grating before such interpretations can be accepted. 29. V. P. Zrelov, J. Ruzicka, A. A. Tyapkin, JINR Rapid Commun. 1[87]-98, 23 (1998). 30. V. P. Zrelor, J. Ruzicka, A. A. Tyapkin, CERN Courier 38 (no. 9), 7 (1998)]. 31. We are grateful to C. Aku-Leh for assistance with the calculations and to D. G. Steel for critical reading of the manuscript. T.E.S. would like to thank the Max￾Planck-Institut fu¨r Festko¨rpertorschung for warm hospitality. Work supported by the NSF under grant DMR 9876862 and by the Air Force Office of Scien￾tific Research under contract F49620-00-1-0328 through the Multidisciplinary University Research Ini￾tiative program. 29 September 2000; accepted 30 November 2000 Directed Assembly of One-Dimensional Nanostructures into Functional Networks Yu Huang,1 * Xiangfeng Duan,1 * Qingqiao Wei,1 Charles M. Lieber1,2† One-dimensional nanostructures, such as nanowires and nanotubes, represent the smallest dimension for efficient transport of electrons and excitons and thus are ideal building blocks for hierarchical assembly of functional nanoscale electronic and photonic structures. We report an approach for the hierarchical assembly of one-dimensional nanostructures into well-defined functional net￾works. We show that nanowires can be assembled into parallel arrays with control of the average separation and, by combining fluidic alignment with surface-patterning techniques, that it is also possible to control periodicity. In addition, complex crossed nanowire arrays can be prepared with layer-by-layer assembly with different flow directions for sequential steps. Transport studies show that the crossed nanowire arrays form electrically conducting networks, with individually addressable device function at each cross point. Nanoscale materials, for example, nanoclus￾ters and nanowires (NWs), represent attrac￾tive building blocks for hierarchical assembly of functional nanoscale devices that could overcome fundamental and economic limita￾tions of conventional lithography-based fab￾rication (1–4). Research focused on zero￾dimensional nanoclusters has led to substan￾tial advances, including the assembly of ar￾rays with order extending from nanometer to micrometer length scales (4–9). In contrast, the assembly of one-dimensional (1D) nano￾R EPORTS 630 26 JANUARY 2001 VOL 291 SCIENCE www.sciencemag.org