Abstract

We have synthesized ordered carbon nanotube (CNT) arrays in porous anodic alumina (PAA) matrix, and have characterized their total optical reflectance and bi-directional reflectance distribution function after each processing step of the microwave plasma chemical vapor deposition process (MPCVD). For a PAA sample without CNT growth, the reflectance shows an oscillating pattern with wavelength that agrees reasonably with a multilayer model. During the MPCVD process, heating the sample significantly reduces the reflectance by 30-40%, the plasma treatment reduces the reflectance by another 5-10%, and the CNT growth further reduces the reflectance by 2-3%. After an atomic layer deposition (ALD) process, the reflectance increases to the embedded CNT arrays. After etching and exposure of CNT tips, the reflectance almost returns to the original pattern with slightly higher reflectance. Bi-directional reflectance distribution function (BRDF) measurements show that the CNT-PAA surface is quite specular as indicated by a large lobe at the specular angle, while the secondary lobe can be attributed to surface roughness.

© 2013 OSA

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2010 (2)

A. H. Wang, P. F. Hsu, and J. J. Cai, “Modeling bidirectional reflection distribution function of microscale random rough surfaces,” Journal of Central South University of Technology 17(2), 228–234 (2010).
[Crossref]

H. Bao, X. L. Ruan, and T. S. Fisher, “Optical properties of ordered vertical arrays of multi-walled carbon nanotubes from FDTD simulations,” Opt. Express 18(6), 6347–6359 (2010).
[Crossref] [PubMed]

2009 (3)

X. J. Wang, J. D. Flicker, B. J. Lee, W. J. Ready, and Z. M. Zhang, “Visible and near-infrared radiative properties of vertically aligned multi-walled carbon nanotubes,” Nanotechnology 20(21), 215704 (2009).
[Crossref] [PubMed]

A. D. Franklin, R. A. Sayer, T. D. Sands, D. B. Janes, and T. S. Fisher, “Vertical Carbon Nanotube Devices With Nanoscale Lengths Controlled Without Lithography,” IEEE Trans. NanoTechnol. 8(4), 469–476 (2009).
[Crossref]

E. Lidorikis and A. C. Ferrari, “Photonics with multiwall carbon nanotube arrays,” ACS Nano 3(5), 1238–1248 (2009).
[Crossref] [PubMed]

2008 (3)

S. Shoji, H. Suzuki, R. P. Zaccaria, Z. Sekkat, and S. Kawata, “Optical polarizer made of uniaxially aligned short single-wall carbon nanotubes embedded in a polymer film,” Phys. Rev. B 77(15), 153407 (2008).
[Crossref]

Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8(2), 446–451 (2008).
[Crossref] [PubMed]

A. D. Franklin, D. B. Janes, J. C. Claussen, T. S. Fisher, and T. D. Sands, “Independently addressable fields of porous anodic alumina embedded in SiO2 on Si,” Appl. Phys. Lett. 92(1), 013122 (2008).
[Crossref]

2007 (3)

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

M. R. Maschmann, A. D. Franklin, T. D. Sands, and T. S. Fisher, “Optimization of carbon nanotube synthesis from porous anodic Al-Fe-Al templates,” Carbon 45(11), 2290–2296 (2007).
[Crossref]

2006 (2)

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

M. R. Maschmann, A. D. Franklin, A. Scott, D. B. Janes, T. D. Sands, and T. S. Fisher, “Lithography-free in situ Pd contacts to templated single-walled carbon nanotubes,” Nano Lett. 6(12), 2712–2717 (2006).
[Crossref] [PubMed]

2005 (3)

H. Y. Jung, J. Kim, J. Hahn, and J. S. Suh, “Well-ordered semiconducting linearly joined carbon nanotube devices at room temperature,” Chem. Phys. Lett. 402(4-6), 535–538 (2005).
[Crossref]

G. F. Zhong, T. Iwasaki, K. Honda, Y. Furukawa, I. Ohdomari, and H. Kawarada, “Very high yield growth of vertically aligned single-walled carbon nanotubes by point-arc microwave plasma CVD,” Chemical Vapor Deposition 11(3), 127–130 (2005).
[Crossref]

S. Hofmann, G. Csányi, A. C. Ferrari, M. C. Payne, and J. Robertson, “Surface diffusion: The low activation energy path for nanotube growth,” Phys. Rev. Lett. 95(3), 036101 (2005).
[Crossref] [PubMed]

2004 (4)

S. Fournier-Bidoz, V. Kitaev, D. Routkevitch, I. Manners, and G. A. Ozin, “Highly ordered nanosphere imprinted nanochannel alumina (NINA),” Adv. Mater. 16(23-24), 2193–2196 (2004).
[Crossref]

P. L. Chen, J. K. Chang, C. T. Kuo, and F. M. Pan, “Anodic aluminum oxide template assisted growth of vertically aligned carbon nanotube arrays by ECR-CVD,” Diamond Related Materials 13(11-12), 1949–1953 (2004).
[Crossref]

S. H. Jeong, H. Y. Hwang, S. K. Hwang, and K. H. Lee, “Carbon nanotubes based on anodic aluminum oxide nano-template,” Carbon 42(10), 2073–2080 (2004).
[Crossref]

J. H. Yen, I. C. Leu, M. T. Wu, C. C. Lin, and M. H. Hon, “Density control for carbon nanotube arrays synthesized by ICP-CVD using AAO/Si as a nanotemplate,” Electrochem. Solid-State Lett. 7(8), H29–H31 (2004).
[Crossref]

2003 (2)

M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
[Crossref]

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

2002 (3)

J. S. Lee and J. S. Suh, “Uniform field emission from aligned carbon nanotubes prepared by CO disproportionation,” J. Appl. Phys. 92(12), 7519–7522 (2002).
[Crossref]

E. C. Walter, K. Ng, M. P. Zach, R. M. Penner, and F. Favier, “Electronic devices from electrodeposited metal nanowires,” Microelectron. Eng. 61–62, 555–561 (2002).
[Crossref]

Z. J. Sun and H. K. Kim, “Growth of ordered, single-domain, alumina nanopore arrays with holographically patterned aluminum films,” Appl. Phys. Lett. 81(18), 3458–3460 (2002).
[Crossref]

2001 (1)

H. Asoh, K. Nishio, M. Nakao, A. Yokoo, T. Tamamura, and H. Masuda, “Fabrication of ideally ordered anodic porous alumina with 63 nm hole periodicity using sulfuric acid,” J. Vac. Sci. Technol. B 19(2), 569–572 (2001).
[Crossref]

1998 (2)

A. P. Li, F. Muller, A. Birner, K. Nielsch, and U. Gosele, “Hexagonal pore arrays with a 50-420 nm interpore distance formed by self-organization in anodic alumina,” J. Appl. Phys. 84(11), 6023–6026 (1998).
[Crossref]

O. Jessensky, F. Muller, and U. Gosele, “Self-organized formation of hexagonal pore arrays in anodic alumina,” Appl. Phys. Lett. 72(10), 1173–1175 (1998).
[Crossref]

1997 (2)

H. Masuda, F. Hasegwa, and S. Ono, “Self-ordering of cell arrangement of anodic porous alumina formed in sulfuric acid solution,” J. Electrochem. Soc. 144(5), L127–L130 (1997).
[Crossref]

H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
[Crossref]

1991 (1)

S. Iijima, “Helical microtubules of graphitic carbon,” Nature 354(6348), 56–58 (1991).
[Crossref]

1967 (1)

1953 (1)

F. Keller, M. S. Hunter, and D. L. Robinson, “Structural features of oxide coatings on aluminium,” J. Electrochem. Soc. 100(9), 411–419 (1953).
[Crossref]

1944 (1)

J. D. Edwards and T. Keller, “The structure of anodic oxide coatings,” Transactions of the American Institute of Mining and Metallurgical Engineers 156, 288–299 (1944).

Ajayan, P. M.

Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8(2), 446–451 (2008).
[Crossref] [PubMed]

Amama, P. B.

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

Asoh, H.

H. Asoh, K. Nishio, M. Nakao, A. Yokoo, T. Tamamura, and H. Masuda, “Fabrication of ideally ordered anodic porous alumina with 63 nm hole periodicity using sulfuric acid,” J. Vac. Sci. Technol. B 19(2), 569–572 (2001).
[Crossref]

H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
[Crossref]

Bao, H.

Benham, G.

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

Bergström, D.

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

Birner, A.

A. P. Li, F. Muller, A. Birner, K. Nielsch, and U. Gosele, “Hexagonal pore arrays with a 50-420 nm interpore distance formed by self-organization in anodic alumina,” J. Appl. Phys. 84(11), 6023–6026 (1998).
[Crossref]

Bur, J. A.

Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8(2), 446–451 (2008).
[Crossref] [PubMed]

Cai, J. J.

A. H. Wang, P. F. Hsu, and J. J. Cai, “Modeling bidirectional reflection distribution function of microscale random rough surfaces,” Journal of Central South University of Technology 17(2), 228–234 (2010).
[Crossref]

Carlson, J.

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

Carnahan, D. L.

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Chang, J. K.

P. L. Chen, J. K. Chang, C. T. Kuo, and F. M. Pan, “Anodic aluminum oxide template assisted growth of vertically aligned carbon nanotube arrays by ECR-CVD,” Diamond Related Materials 13(11-12), 1949–1953 (2004).
[Crossref]

Chen, P. L.

P. L. Chen, J. K. Chang, C. T. Kuo, and F. M. Pan, “Anodic aluminum oxide template assisted growth of vertically aligned carbon nanotube arrays by ECR-CVD,” Diamond Related Materials 13(11-12), 1949–1953 (2004).
[Crossref]

Choi, J. H.

M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
[Crossref]

Ci, L. J.

Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8(2), 446–451 (2008).
[Crossref] [PubMed]

Claussen, J. C.

A. D. Franklin, D. B. Janes, J. C. Claussen, T. S. Fisher, and T. D. Sands, “Independently addressable fields of porous anodic alumina embedded in SiO2 on Si,” Appl. Phys. Lett. 92(1), 013122 (2008).
[Crossref]

Csányi, G.

S. Hofmann, G. Csányi, A. C. Ferrari, M. C. Payne, and J. Robertson, “Surface diffusion: The low activation energy path for nanotube growth,” Phys. Rev. Lett. 95(3), 036101 (2005).
[Crossref] [PubMed]

Edwards, J. D.

J. D. Edwards and T. Keller, “The structure of anodic oxide coatings,” Transactions of the American Institute of Mining and Metallurgical Engineers 156, 288–299 (1944).

Favier, F.

E. C. Walter, K. Ng, M. P. Zach, R. M. Penner, and F. Favier, “Electronic devices from electrodeposited metal nanowires,” Microelectron. Eng. 61–62, 555–561 (2002).
[Crossref]

Ferrari, A. C.

E. Lidorikis and A. C. Ferrari, “Photonics with multiwall carbon nanotube arrays,” ACS Nano 3(5), 1238–1248 (2009).
[Crossref] [PubMed]

S. Hofmann, G. Csányi, A. C. Ferrari, M. C. Payne, and J. Robertson, “Surface diffusion: The low activation energy path for nanotube growth,” Phys. Rev. Lett. 95(3), 036101 (2005).
[Crossref] [PubMed]

Fisher, T. S.

H. Bao, X. L. Ruan, and T. S. Fisher, “Optical properties of ordered vertical arrays of multi-walled carbon nanotubes from FDTD simulations,” Opt. Express 18(6), 6347–6359 (2010).
[Crossref] [PubMed]

A. D. Franklin, R. A. Sayer, T. D. Sands, D. B. Janes, and T. S. Fisher, “Vertical Carbon Nanotube Devices With Nanoscale Lengths Controlled Without Lithography,” IEEE Trans. NanoTechnol. 8(4), 469–476 (2009).
[Crossref]

A. D. Franklin, D. B. Janes, J. C. Claussen, T. S. Fisher, and T. D. Sands, “Independently addressable fields of porous anodic alumina embedded in SiO2 on Si,” Appl. Phys. Lett. 92(1), 013122 (2008).
[Crossref]

M. R. Maschmann, A. D. Franklin, T. D. Sands, and T. S. Fisher, “Optimization of carbon nanotube synthesis from porous anodic Al-Fe-Al templates,” Carbon 45(11), 2290–2296 (2007).
[Crossref]

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

M. R. Maschmann, A. D. Franklin, A. Scott, D. B. Janes, T. D. Sands, and T. S. Fisher, “Lithography-free in situ Pd contacts to templated single-walled carbon nanotubes,” Nano Lett. 6(12), 2712–2717 (2006).
[Crossref] [PubMed]

Flicker, J. D.

X. J. Wang, J. D. Flicker, B. J. Lee, W. J. Ready, and Z. M. Zhang, “Visible and near-infrared radiative properties of vertically aligned multi-walled carbon nanotubes,” Nanotechnology 20(21), 215704 (2009).
[Crossref] [PubMed]

Fournier-Bidoz, S.

S. Fournier-Bidoz, V. Kitaev, D. Routkevitch, I. Manners, and G. A. Ozin, “Highly ordered nanosphere imprinted nanochannel alumina (NINA),” Adv. Mater. 16(23-24), 2193–2196 (2004).
[Crossref]

Franklin, A. D.

A. D. Franklin, R. A. Sayer, T. D. Sands, D. B. Janes, and T. S. Fisher, “Vertical Carbon Nanotube Devices With Nanoscale Lengths Controlled Without Lithography,” IEEE Trans. NanoTechnol. 8(4), 469–476 (2009).
[Crossref]

A. D. Franklin, D. B. Janes, J. C. Claussen, T. S. Fisher, and T. D. Sands, “Independently addressable fields of porous anodic alumina embedded in SiO2 on Si,” Appl. Phys. Lett. 92(1), 013122 (2008).
[Crossref]

M. R. Maschmann, A. D. Franklin, T. D. Sands, and T. S. Fisher, “Optimization of carbon nanotube synthesis from porous anodic Al-Fe-Al templates,” Carbon 45(11), 2290–2296 (2007).
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M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
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K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
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M. R. Maschmann, A. D. Franklin, T. D. Sands, and T. S. Fisher, “Optimization of carbon nanotube synthesis from porous anodic Al-Fe-Al templates,” Carbon 45(11), 2290–2296 (2007).
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M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

M. R. Maschmann, A. D. Franklin, A. Scott, D. B. Janes, T. D. Sands, and T. S. Fisher, “Lithography-free in situ Pd contacts to templated single-walled carbon nanotubes,” Nano Lett. 6(12), 2712–2717 (2006).
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O. Jessensky, F. Muller, and U. Gosele, “Self-organized formation of hexagonal pore arrays in anodic alumina,” Appl. Phys. Lett. 72(10), 1173–1175 (1998).
[Crossref]

A. P. Li, F. Muller, A. Birner, K. Nielsch, and U. Gosele, “Hexagonal pore arrays with a 50-420 nm interpore distance formed by self-organization in anodic alumina,” J. Appl. Phys. 84(11), 6023–6026 (1998).
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H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
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H. Asoh, K. Nishio, M. Nakao, A. Yokoo, T. Tamamura, and H. Masuda, “Fabrication of ideally ordered anodic porous alumina with 63 nm hole periodicity using sulfuric acid,” J. Vac. Sci. Technol. B 19(2), 569–572 (2001).
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G. F. Zhong, T. Iwasaki, K. Honda, Y. Furukawa, I. Ohdomari, and H. Kawarada, “Very high yield growth of vertically aligned single-walled carbon nanotubes by point-arc microwave plasma CVD,” Chemical Vapor Deposition 11(3), 127–130 (2005).
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S. Fournier-Bidoz, V. Kitaev, D. Routkevitch, I. Manners, and G. A. Ozin, “Highly ordered nanosphere imprinted nanochannel alumina (NINA),” Adv. Mater. 16(23-24), 2193–2196 (2004).
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P. L. Chen, J. K. Chang, C. T. Kuo, and F. M. Pan, “Anodic aluminum oxide template assisted growth of vertically aligned carbon nanotube arrays by ECR-CVD,” Diamond Related Materials 13(11-12), 1949–1953 (2004).
[Crossref]

Park, C. Y.

M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
[Crossref]

Park, J. B.

M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
[Crossref]

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S. Hofmann, G. Csányi, A. C. Ferrari, M. C. Payne, and J. Robertson, “Surface diffusion: The low activation energy path for nanotube growth,” Phys. Rev. Lett. 95(3), 036101 (2005).
[Crossref] [PubMed]

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E. C. Walter, K. Ng, M. P. Zach, R. M. Penner, and F. Favier, “Electronic devices from electrodeposited metal nanowires,” Microelectron. Eng. 61–62, 555–561 (2002).
[Crossref]

Powell, J.

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

Rao, D. V. G. L. N.

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Ready, W. J.

X. J. Wang, J. D. Flicker, B. J. Lee, W. J. Ready, and Z. M. Zhang, “Visible and near-infrared radiative properties of vertically aligned multi-walled carbon nanotubes,” Nanotechnology 20(21), 215704 (2009).
[Crossref] [PubMed]

Ren, Z. F.

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
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S. Hofmann, G. Csányi, A. C. Ferrari, M. C. Payne, and J. Robertson, “Surface diffusion: The low activation energy path for nanotube growth,” Phys. Rev. Lett. 95(3), 036101 (2005).
[Crossref] [PubMed]

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F. Keller, M. S. Hunter, and D. L. Robinson, “Structural features of oxide coatings on aluminium,” J. Electrochem. Soc. 100(9), 411–419 (1953).
[Crossref]

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S. Fournier-Bidoz, V. Kitaev, D. Routkevitch, I. Manners, and G. A. Ozin, “Highly ordered nanosphere imprinted nanochannel alumina (NINA),” Adv. Mater. 16(23-24), 2193–2196 (2004).
[Crossref]

Ruan, X. L.

Rybczynski, J.

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Sands, T. D.

A. D. Franklin, R. A. Sayer, T. D. Sands, D. B. Janes, and T. S. Fisher, “Vertical Carbon Nanotube Devices With Nanoscale Lengths Controlled Without Lithography,” IEEE Trans. NanoTechnol. 8(4), 469–476 (2009).
[Crossref]

A. D. Franklin, D. B. Janes, J. C. Claussen, T. S. Fisher, and T. D. Sands, “Independently addressable fields of porous anodic alumina embedded in SiO2 on Si,” Appl. Phys. Lett. 92(1), 013122 (2008).
[Crossref]

M. R. Maschmann, A. D. Franklin, T. D. Sands, and T. S. Fisher, “Optimization of carbon nanotube synthesis from porous anodic Al-Fe-Al templates,” Carbon 45(11), 2290–2296 (2007).
[Crossref]

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

M. R. Maschmann, A. D. Franklin, A. Scott, D. B. Janes, T. D. Sands, and T. S. Fisher, “Lithography-free in situ Pd contacts to templated single-walled carbon nanotubes,” Nano Lett. 6(12), 2712–2717 (2006).
[Crossref] [PubMed]

Satoh, M.

H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
[Crossref]

Sayer, R. A.

A. D. Franklin, R. A. Sayer, T. D. Sands, D. B. Janes, and T. S. Fisher, “Vertical Carbon Nanotube Devices With Nanoscale Lengths Controlled Without Lithography,” IEEE Trans. NanoTechnol. 8(4), 469–476 (2009).
[Crossref]

Scott, A.

M. R. Maschmann, A. D. Franklin, A. Scott, D. B. Janes, T. D. Sands, and T. S. Fisher, “Lithography-free in situ Pd contacts to templated single-walled carbon nanotubes,” Nano Lett. 6(12), 2712–2717 (2006).
[Crossref] [PubMed]

Sekkat, Z.

S. Shoji, H. Suzuki, R. P. Zaccaria, Z. Sekkat, and S. Kawata, “Optical polarizer made of uniaxially aligned short single-wall carbon nanotubes embedded in a polymer film,” Phys. Rev. B 77(15), 153407 (2008).
[Crossref]

Sennett, M.

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Shoji, S.

S. Shoji, H. Suzuki, R. P. Zaccaria, Z. Sekkat, and S. Kawata, “Optical polarizer made of uniaxially aligned short single-wall carbon nanotubes embedded in a polymer film,” Phys. Rev. B 77(15), 153407 (2008).
[Crossref]

Sparrow, E. M.

Stach, E. A.

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

Steeves, D.

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Suh, J. S.

H. Y. Jung, J. Kim, J. Hahn, and J. S. Suh, “Well-ordered semiconducting linearly joined carbon nanotube devices at room temperature,” Chem. Phys. Lett. 402(4-6), 535–538 (2005).
[Crossref]

J. S. Lee and J. S. Suh, “Uniform field emission from aligned carbon nanotubes prepared by CO disproportionation,” J. Appl. Phys. 92(12), 7519–7522 (2002).
[Crossref]

Sun, Z. J.

Z. J. Sun and H. K. Kim, “Growth of ordered, single-domain, alumina nanopore arrays with holographically patterned aluminum films,” Appl. Phys. Lett. 81(18), 3458–3460 (2002).
[Crossref]

Suzuki, H.

S. Shoji, H. Suzuki, R. P. Zaccaria, Z. Sekkat, and S. Kawata, “Optical polarizer made of uniaxially aligned short single-wall carbon nanotubes embedded in a polymer film,” Phys. Rev. B 77(15), 153407 (2008).
[Crossref]

Tamamura, T.

H. Asoh, K. Nishio, M. Nakao, A. Yokoo, T. Tamamura, and H. Masuda, “Fabrication of ideally ordered anodic porous alumina with 63 nm hole periodicity using sulfuric acid,” J. Vac. Sci. Technol. B 19(2), 569–572 (2001).
[Crossref]

H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
[Crossref]

Torrance, K. E.

Vidan, A.

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

Walter, E. C.

E. C. Walter, K. Ng, M. P. Zach, R. M. Penner, and F. Favier, “Electronic devices from electrodeposited metal nanowires,” Microelectron. Eng. 61–62, 555–561 (2002).
[Crossref]

Wang, A. H.

A. H. Wang, P. F. Hsu, and J. J. Cai, “Modeling bidirectional reflection distribution function of microscale random rough surfaces,” Journal of Central South University of Technology 17(2), 228–234 (2010).
[Crossref]

Wang, D. Z.

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Wang, X. J.

X. J. Wang, J. D. Flicker, B. J. Lee, W. J. Ready, and Z. M. Zhang, “Visible and near-infrared radiative properties of vertically aligned multi-walled carbon nanotubes,” Nanotechnology 20(21), 215704 (2009).
[Crossref] [PubMed]

Wang, Y.

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

Wu, M. T.

J. H. Yen, I. C. Leu, M. T. Wu, C. C. Lin, and M. H. Hon, “Density control for carbon nanotube arrays synthesized by ICP-CVD using AAO/Si as a nanotemplate,” Electrochem. Solid-State Lett. 7(8), H29–H31 (2004).
[Crossref]

Wu, P. F.

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Yamada, H.

H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
[Crossref]

Yang, Z. P.

Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8(2), 446–451 (2008).
[Crossref] [PubMed]

Yen, J. H.

J. H. Yen, I. C. Leu, M. T. Wu, C. C. Lin, and M. H. Hon, “Density control for carbon nanotube arrays synthesized by ICP-CVD using AAO/Si as a nanotemplate,” Electrochem. Solid-State Lett. 7(8), H29–H31 (2004).
[Crossref]

Yokoo, A.

H. Asoh, K. Nishio, M. Nakao, A. Yokoo, T. Tamamura, and H. Masuda, “Fabrication of ideally ordered anodic porous alumina with 63 nm hole periodicity using sulfuric acid,” J. Vac. Sci. Technol. B 19(2), 569–572 (2001).
[Crossref]

Yoo, J. B.

M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
[Crossref]

Zaccaria, R. P.

S. Shoji, H. Suzuki, R. P. Zaccaria, Z. Sekkat, and S. Kawata, “Optical polarizer made of uniaxially aligned short single-wall carbon nanotubes embedded in a polymer film,” Phys. Rev. B 77(15), 153407 (2008).
[Crossref]

Zach, M. P.

E. C. Walter, K. Ng, M. P. Zach, R. M. Penner, and F. Favier, “Electronic devices from electrodeposited metal nanowires,” Microelectron. Eng. 61–62, 555–561 (2002).
[Crossref]

Zakharov, D. N.

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

Zhang, Z. M.

X. J. Wang, J. D. Flicker, B. J. Lee, W. J. Ready, and Z. M. Zhang, “Visible and near-infrared radiative properties of vertically aligned multi-walled carbon nanotubes,” Nanotechnology 20(21), 215704 (2009).
[Crossref] [PubMed]

Zhong, G. F.

G. F. Zhong, T. Iwasaki, K. Honda, Y. Furukawa, I. Ohdomari, and H. Kawarada, “Very high yield growth of vertically aligned single-walled carbon nanotubes by point-arc microwave plasma CVD,” Chemical Vapor Deposition 11(3), 127–130 (2005).
[Crossref]

ACS Nano (1)

E. Lidorikis and A. C. Ferrari, “Photonics with multiwall carbon nanotube arrays,” ACS Nano 3(5), 1238–1248 (2009).
[Crossref] [PubMed]

Adv. Mater. (2)

K. Kempa, J. Rybczynski, Z. P. Huang, K. Gregorczyk, A. Vidan, B. Kimball, J. Carlson, G. Benham, Y. Wang, A. Herczynski, and Z. F. Ren, “Carbon nanotubes as optical antennae,” Adv. Mater. 19(3), 421–426 (2007).
[Crossref]

S. Fournier-Bidoz, V. Kitaev, D. Routkevitch, I. Manners, and G. A. Ozin, “Highly ordered nanosphere imprinted nanochannel alumina (NINA),” Adv. Mater. 16(23-24), 2193–2196 (2004).
[Crossref]

Appl. Phys. Lett. (4)

A. D. Franklin, D. B. Janes, J. C. Claussen, T. S. Fisher, and T. D. Sands, “Independently addressable fields of porous anodic alumina embedded in SiO2 on Si,” Appl. Phys. Lett. 92(1), 013122 (2008).
[Crossref]

H. Masuda, H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, “Highly ordered nanochannel-array architecture in anodic alumina,” Appl. Phys. Lett. 71(19), 2770–2772 (1997).
[Crossref]

O. Jessensky, F. Muller, and U. Gosele, “Self-organized formation of hexagonal pore arrays in anodic alumina,” Appl. Phys. Lett. 72(10), 1173–1175 (1998).
[Crossref]

Z. J. Sun and H. K. Kim, “Growth of ordered, single-domain, alumina nanopore arrays with holographically patterned aluminum films,” Appl. Phys. Lett. 81(18), 3458–3460 (2002).
[Crossref]

Carbon (2)

M. R. Maschmann, A. D. Franklin, T. D. Sands, and T. S. Fisher, “Optimization of carbon nanotube synthesis from porous anodic Al-Fe-Al templates,” Carbon 45(11), 2290–2296 (2007).
[Crossref]

S. H. Jeong, H. Y. Hwang, S. K. Hwang, and K. H. Lee, “Carbon nanotubes based on anodic aluminum oxide nano-template,” Carbon 42(10), 2073–2080 (2004).
[Crossref]

Chem. Phys. Lett. (1)

H. Y. Jung, J. Kim, J. Hahn, and J. S. Suh, “Well-ordered semiconducting linearly joined carbon nanotube devices at room temperature,” Chem. Phys. Lett. 402(4-6), 535–538 (2005).
[Crossref]

Chemical Vapor Deposition (1)

G. F. Zhong, T. Iwasaki, K. Honda, Y. Furukawa, I. Ohdomari, and H. Kawarada, “Very high yield growth of vertically aligned single-walled carbon nanotubes by point-arc microwave plasma CVD,” Chemical Vapor Deposition 11(3), 127–130 (2005).
[Crossref]

Diamond Related Materials (2)

P. L. Chen, J. K. Chang, C. T. Kuo, and F. M. Pan, “Anodic aluminum oxide template assisted growth of vertically aligned carbon nanotube arrays by ECR-CVD,” Diamond Related Materials 13(11-12), 1949–1953 (2004).
[Crossref]

M. J. Kim, T. Y. Lee, J. H. Choi, J. B. Park, J. S. Lee, S. K. Kim, J. B. Yoo, and C. Y. Park, “Growth of carbon nanotubes with anodic aluminum oxide formed on the catalytic metal-coated Si substrate,” Diamond Related Materials 12(3-7), 870–873 (2003).
[Crossref]

Electrochem. Solid-State Lett. (1)

J. H. Yen, I. C. Leu, M. T. Wu, C. C. Lin, and M. H. Hon, “Density control for carbon nanotube arrays synthesized by ICP-CVD using AAO/Si as a nanotemplate,” Electrochem. Solid-State Lett. 7(8), H29–H31 (2004).
[Crossref]

IEEE Trans. NanoTechnol. (1)

A. D. Franklin, R. A. Sayer, T. D. Sands, D. B. Janes, and T. S. Fisher, “Vertical Carbon Nanotube Devices With Nanoscale Lengths Controlled Without Lithography,” IEEE Trans. NanoTechnol. 8(4), 469–476 (2009).
[Crossref]

J. Appl. Phys. (3)

D. Bergström, J. Powell, and A. F. H. Kaplan, “A ray-tracing analysis of the absorption of light by smooth and rough metal surfaces,” J. Appl. Phys. 101(11), 113504 (2007).
[Crossref]

A. P. Li, F. Muller, A. Birner, K. Nielsch, and U. Gosele, “Hexagonal pore arrays with a 50-420 nm interpore distance formed by self-organization in anodic alumina,” J. Appl. Phys. 84(11), 6023–6026 (1998).
[Crossref]

J. S. Lee and J. S. Suh, “Uniform field emission from aligned carbon nanotubes prepared by CO disproportionation,” J. Appl. Phys. 92(12), 7519–7522 (2002).
[Crossref]

J. Electrochem. Soc. (2)

H. Masuda, F. Hasegwa, and S. Ono, “Self-ordering of cell arrangement of anodic porous alumina formed in sulfuric acid solution,” J. Electrochem. Soc. 144(5), L127–L130 (1997).
[Crossref]

F. Keller, M. S. Hunter, and D. L. Robinson, “Structural features of oxide coatings on aluminium,” J. Electrochem. Soc. 100(9), 411–419 (1953).
[Crossref]

J. Opt. Soc. Am. (1)

J. Vac. Sci. Technol. B (1)

H. Asoh, K. Nishio, M. Nakao, A. Yokoo, T. Tamamura, and H. Masuda, “Fabrication of ideally ordered anodic porous alumina with 63 nm hole periodicity using sulfuric acid,” J. Vac. Sci. Technol. B 19(2), 569–572 (2001).
[Crossref]

Journal of Central South University of Technology (1)

A. H. Wang, P. F. Hsu, and J. J. Cai, “Modeling bidirectional reflection distribution function of microscale random rough surfaces,” Journal of Central South University of Technology 17(2), 228–234 (2010).
[Crossref]

Microelectron. Eng. (1)

E. C. Walter, K. Ng, M. P. Zach, R. M. Penner, and F. Favier, “Electronic devices from electrodeposited metal nanowires,” Microelectron. Eng. 61–62, 555–561 (2002).
[Crossref]

Nano Lett. (3)

M. R. Maschmann, A. D. Franklin, A. Scott, D. B. Janes, T. D. Sands, and T. S. Fisher, “Lithography-free in situ Pd contacts to templated single-walled carbon nanotubes,” Nano Lett. 6(12), 2712–2717 (2006).
[Crossref] [PubMed]

K. Kempa, B. Kimball, J. Rybczynski, Z. P. Huang, P. F. Wu, D. Steeves, M. Sennett, M. Giersig, D. V. G. L. N. Rao, D. L. Carnahan, D. Z. Wang, J. Y. Lao, W. Z. Li, and Z. F. Ren, “Photonic crystals based on periodic arrays of aligned carbon nanotubes,” Nano Lett. 3(1), 13–18 (2003).
[Crossref]

Z. P. Yang, L. J. Ci, J. A. Bur, S. Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8(2), 446–451 (2008).
[Crossref] [PubMed]

Nanotechnology (2)

X. J. Wang, J. D. Flicker, B. J. Lee, W. J. Ready, and Z. M. Zhang, “Visible and near-infrared radiative properties of vertically aligned multi-walled carbon nanotubes,” Nanotechnology 20(21), 215704 (2009).
[Crossref] [PubMed]

M. R. Maschmann, A. D. Franklin, P. B. Amama, D. N. Zakharov, E. A. Stach, T. D. Sands, and T. S. Fisher, “Vertical single- and double-walled carbon nanotubes grown from modified porous anodic alumina templates,” Nanotechnology 17(15), 3925–3929 (2006).
[Crossref]

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S. Iijima, “Helical microtubules of graphitic carbon,” Nature 354(6348), 56–58 (1991).
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Phys. Rev. B (1)

S. Shoji, H. Suzuki, R. P. Zaccaria, Z. Sekkat, and S. Kawata, “Optical polarizer made of uniaxially aligned short single-wall carbon nanotubes embedded in a polymer film,” Phys. Rev. B 77(15), 153407 (2008).
[Crossref]

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S. Hofmann, G. Csányi, A. C. Ferrari, M. C. Payne, and J. Robertson, “Surface diffusion: The low activation energy path for nanotube growth,” Phys. Rev. Lett. 95(3), 036101 (2005).
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Figures (9)

Fig. 1
Fig. 1

Cross-sectional schematic of: (a) metal film stack deposited on Si wafer, (b) PAA film after anodization, and (c) SWCNTs emerging from PAA pores after undergoing MPCVD synthesis; the box highlights the vertical SWCNT channels.

Fig. 2
Fig. 2

Cross-sectional SEM images of (a) SWCNTs growing vertically from PAA and (b) SWCNTs draping along the PAA surface and appearing as bright strands.

Fig. 3
Fig. 3

(a) CNTs in PAA sample after ALD deposition of alumina; (b) CNTs in PAA sample after ALD deposition and final etches.

Fig. 4
Fig. 4

(a) PAA sample with CNT growth and alumina coating from an ALD process, with a transparent color overlay on the right side to indicate the different parts of the metal stack. The iron layer was coated by the alumina. (b) Top view of PAA sample with CNT growth and alumina coating from an ALD process. (c) An RIE etch to expose CNT tips, and some examples of tips are circled. (d) Zoomed in view of a CNT tip on the surface of the sample after anodization, CNT growth, ALD process of alumina, and final etches to obtain CNT tips. The tip is circled.

Fig. 5
Fig. 5

(a) A sketch of integrating sphere measurement. (b) The actual experimental set-up.

Fig. 6
Fig. 6

A comparison between the theoretically calculated reflectance at 8 degrees and the measured reflectance for a 750 nm thick PAA sample.

Fig. 7
Fig. 7

Reflectance measurements at 8 degrees of the three pieces of PAA broken from the same original anodized piece and allowed to complete different steps of the CVD process: heating, plasma treatment, and CNT growth.

Fig. 8
Fig. 8

A comparison of total reflectance at 8 degrees after each of the four steps in the full process. PAA: after PAA construction. Growth: after CNT growth. ALD: after the ALD process. Tips: after etching to expose the CNT tips.

Fig. 9
Fig. 9

(a) BRDF measurements of the sample with CNT tips poking out of an alumina substrate as a function of angle. The legend indicates the angle of incidence. (b) A magnified region of the BRDF at 35° incidence angle to highlight the secondary lobe.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

δ p =λ/ 4πκ ,
r 12 = n 1 cos θ 1 - n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 , r 23 = n 2 cos θ 2 - n 3 cos θ 3 n 2 cos θ 2 + n 3 cos θ 3 ,
n 1 sin θ 1 = n 2 sin θ 2 = n 3 sin θ 3 .
r= r 12 + r 23 e i2δ 1 r 12 r 23 e i2δ ,whereδ=( 2π λ 0 )  n 2 dcos θ 2 .
R= | r | 2 .

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