Abstract

We report our theoretical investigation onto the shape dependence of two-photon absorption (TPA) in semiconductor nanocrystals (NCs). Based on a four-band model under effective mass approximation, we have developed a simple analytical theory capable of providing a quantitative explanation of the recent TPA measurement on CdS nanorods [Appl. Phys. Lett. 94, 103117 (2009)]. With this theory, we have systematically revealed the characteristics of TPA in CdSe and ZnO NCs with four different shapes: sphere, cube, cylinder and cuboid. Due to the splitting of degenerate energy levels caused by the decreased degree of symmetry, nanocuboids and nanocubes exhibit greater TPA cross-sections than nanocylinders and nanospheres of similar sizes, respectively. Similarly, nanocuboids and nanocylinders possess larger TPA cross-sections than nanocubes and nanospheres of similar lateral dimension, respectively. Given TPA-allowed transitions, nanocuboids show stronger size dependence than nanocylinders. The size dependence of TPA cross-section is more sensitive to the lateral size than the longitudinal size in the cases of nanocylinders and nanocuboids.

© 2009 Optical Society of America

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    [CrossRef]
  38. P. Lawaetz, "Valence-band parameters in cubic semiconductors," Phys. Rev. B 4, 3460-3467 (1971).
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  39. A. L. Efros and A. V. Arondina, "Band-edge absorption and luminescence of nonspherical nanometer-size crystals," Phys. Rev. B 47, 10005-10007 (1993).
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  41. F. Buonocore, D. Ninno, and G. Iadonisi, "Localized states in arbitrarily shaped quantum wire: a variation-perturbation technique," Phys. Stat. Sol.(b) 225, 343-352 (2001).
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  43. G. Cantele, D. Ninno, and G. Iadonisi, "Calculation of the infrared optical transitions in semiconductor ellipsoidal quantum dots," Nano Lett. 1, 121-124 (2001).
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  47. L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, "Band gap variation of size- and shape-controlled colloidal CdSe quantum rods," Nano Lett. 1, 349-351 (2001).
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    [CrossRef] [PubMed]

2009 (2)

X. Li, J. Embden, W. M. Chon, and M. Gu, "Enhanced two-photon absorption of CdS nanocrystal rods," Appl. Phys.Lett. 94, 103117-103119 (2009).
[CrossRef]

T. Yatsui, S. Sangu, K. Kobayashi, T. Kawazoe, M. Ohtsu, J. Yoo, and G. C. Yi, "Nanophotonic energy up conversion using ZnO nanorod double-quantum-well structures," Appl. Phys. Lett. 94, 083113-083115 (2009).
[CrossRef]

2008 (3)

C. Zhang, F. Zhang, S. Qian, N. Kumar, J. Hahm, and J. Xu, "Multiphoton absorption induced amplified spontaneous emission from biocatalyst-synthesized ZnO nanorods," Appl. Phys. Lett. 92, 233116-233118 (2008).
[CrossRef]

G. C. Xing, W. Ji, Y. G. Zheng, and J. Y. Ying, "Two- and three-photon absorption of semiconductor quantum dots in the vicinity of half of lowest exciton energy," Appl. Phys. Lett. 93, 241114-241116 (2008).
[CrossRef]

D. L. Yao, G. Zhang, and B. W. Li, "A universal expression of band gap for silicon nanowires of different cross-section geometries," Nano Lett. 8, 4557-4561 (2008).
[CrossRef]

2007 (7)

G. S. He, K. T. Yong, Q. D. Zheng, Y. Sahoo, A. Baev, A. I. Ryasnyanskiy, and P. N. Prasad, "Multi-photon excitation properties of CdSe quantum dots solutions and optical limiting behavior in infrared range," Opt. Express 15, 12818-12833 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12818.
[CrossRef] [PubMed]

Y. Qu, W. Ji, Y. Zheng, and J. Y. Ying, "Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots," Appl. Phys. Lett. 90, 133112-133114 (2007).
[CrossRef]

C. Gan, M. Xiao, D. Battaglia, N. Pradhan, and X. G Peng, "Size dependence of nonlinear optical absorption and refraction of Mn-doped ZnSe nanocrystals," Appl. Phys. Lett. 91, 201103-201105 (2007).
[CrossRef]

L. Y. Pan, N. Tamai, K. Kamada, and S. Deki, "Nonlinear optical properties of thiol-capped CdTe quantum dots in nonresonant region," Appl. Phys. Lett. 91, 051902-051904 (2007).
[CrossRef]

P. S. Nair, K. P. Fritz, and G. D. Scholes, "Evolutionary shape control during colloidal quantum-dot growth," Small 3, 481-487 (2007).
[CrossRef] [PubMed]

J. Kim, P. S. Nair, C. Y. Wong, and G. D. Scholes, "Sizing up the exciton in complex-shaped semiconductor nanocrystals," Nano Lett. 7, 3884-3890 (2007).
[CrossRef]

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
[CrossRef]

2006 (3)

S. Kumar and T. Nann, "Shape control of II-VI semiconductor nanomaterials," Small 2, 316-329 (2006).
[CrossRef] [PubMed]

P. D. Cozzoli, T. Pellegrino, and L. Manna, "Synthesis, properties and perspectives of hybrid nanocrystal structures," Chem. Soc. Rev. 35, 1195-1208 (2006).
[CrossRef] [PubMed]

J. X. Cao, X. G. Gong, J. X. Zhong, and R. Q. Wu, "Sharp corners in the cross section of ultrathin Si nanowires," Phys. Rev. Lett. 97, 136105-136108 (2006).
[CrossRef] [PubMed]

2005 (7)

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum qots for live cells, in vivo imaging, and diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, "Coherent manipulation of coupled electron spins in semiconductor quantum dots," Science 309, 2180-2184 (2005).
[CrossRef] [PubMed]

Y. Yin and A. P. Alivisatos, "Colloidal nanocrystal synthesis and the organic-inorganic interface," Nature 437, 664-670 (2005).
[CrossRef] [PubMed]

A. G. Kanaras, C. Sonnichsen, H. T. Liu, and A. P. Alivisatos, "Controlled synthesis of hyperbranched inorganic nanocrystals with rich three-dimensional structures," Nano Lett. 5, 2164-2167 (2005).
[CrossRef] [PubMed]

D. Tari, M. De Giorgi, F. Della Sala, L. Carbone, R. Krahne, L. Manna, R. Cingolani, S. Kudera, and W. J. Parak, "Optical properties of tetrapod-shaped CdTe nanocrystals," Appl. Phys. Lett. 87, 224101-224103 (2005).
[CrossRef]

N. L. Thomas, E. Herz, O. Schöps and U. Woggon, "Exciton Fine Structure in Single CdSe Nanorods," Phys. Rev. Lett. 94, 016803-016806 (2005).
[CrossRef] [PubMed]

W. Ji, Y. Qu, J. Mi, Y. Zheng, and J. Y. Ying, "Wavelength scaling for multiphoton absorption in semiconductor quantum dots," Chinese Opt. Lett. 3, S203-S204 (2005).

2004 (2)

V. V. Nikesh, A. Dharmadhikari, H. Ono, S. Nozaki, G. S. Kumar, and S. Mahamuni, "Optical nonlinearity of monodispersed, capped ZnS quantum particles," Appl. Phys. Lett. 84, 46024604-46024606 (2004).
[CrossRef]

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L. W. Wang, and A. P. Alivisatos, "Colloidal nanocrystal heterostructures with linear and branched topology," Nature 430, 190-195 (2004).
[CrossRef] [PubMed]

2003 (4)

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, "Controlled growth of tetrapod-branched inorganic nanocrystals," Nat. Mater. 2, 382-385 (2003).
[CrossRef] [PubMed]

S. M. Lee, S. N. Cho, and J. Cheon, "Anisotropic shape control of colloidal inorganic nanocrystals," Adv. Mater. 15, 441-444 (2003).
[CrossRef]

J. Li and L. W. Wang, "Shape effects on electronic states of nanocrystals," Nano Lett. 3, 1357-1363 (2003).
[CrossRef]

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, "Water-soluble quantum dots for multiphoton fluorescence imaging in vivo," Science 300, 1434-1436 (2003).
[CrossRef] [PubMed]

2002 (1)

B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, "In vivo imaging of quantum dots encapsulated in phospholipid micelles," Science 298, 1759-1762 (2002).
[CrossRef] [PubMed]

2001 (5)

M. B. Mohamed, C. Burda, and M. A. El-Sayed, "Shape dependent ultrafast relaxation dynamics of CdSe nanocrystals: Nanorods vs Nanodots," Nano Lett. 1, 589-593 (2001).
[CrossRef]

L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, "Band gap variation of size- and shape-controlled colloidal CdSe quantum rods," Nano Lett. 1, 349-351 (2001).
[CrossRef]

E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin and J. T. Devreese, "Development of an eight-band theory for quantum dot heterostructures," Phys. Rev. B 64, 245328-245343 (2001).
[CrossRef]

F. Buonocore, D. Ninno, and G. Iadonisi, "Localized states in arbitrarily shaped quantum wire: a variation-perturbation technique," Phys. Stat. Sol.(b) 225, 343-352 (2001).
[CrossRef]

G. Cantele, D. Ninno, and G. Iadonisi, "Calculation of the infrared optical transitions in semiconductor ellipsoidal quantum dots," Nano Lett. 1, 121-124 (2001).
[CrossRef]

2000 (2)

G. Cantele, D. Ninno, and G. Iadonisi, "Confined states in ellipsoidal quantum dots," J. Phys.: Condens. Matter 12, 9019-9036 (2000).
[CrossRef]

J. Staromlynska, T. J. McKay, and P. Wilson, "Broadband optical limiting based on excited state absorption in Pt:ethynyl," J. Appl. Phys. 88, 1726-1732 (2000).
[CrossRef]

1999 (1)

D. Ninno, G. Iadonisi, and F. Buonocore, "Carrier localization and photoluminescence in porous silicon" Solid State Commun. 112, 521-525 (1999).
[CrossRef]

1996 (6)

M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop’ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment," Appl. Phys. Lett. 68, 979-981 (1996).
[CrossRef]

A. V. Fedorov, A. V. Baranov, and K. Inoue, "Two-photon transitions in systems with semiconductor quantum dots," Phys. Rev. B 54, 8627-8632 (1996).
[CrossRef]

M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, "Size-dependent two-photon excitation spectroscopy of CdSe nanocrystals," Phys. Rev. B 53, 12629-12632 (1996).
[CrossRef]

A. P. Alivisatos, "Semiconductor clusters, nanocrystals, and quantum dots," Science 271, 933-937 (1996).
[CrossRef]

D. J. Norris, A. L. Efros, M. Rosen, and M. G. Bawendi, "Size dependence of exciton fine structure in CdSe quantum dots," Phys. Rev. B 53, 16347-16354 (1996).
[CrossRef]

D. J. Norris and M. G. Bawendi, "Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots," Phys.Rev. B 53, 16338-16346 (1996).
[CrossRef]

1993 (2)

A. L. Efros and A. V. Arondina, "Band-edge absorption and luminescence of nonspherical nanometer-size crystals," Phys. Rev. B 47, 10005-10007 (1993).
[CrossRef]

S. L. Goff and B. Stébé, "Influence of longitudinal and lateral confinements on excitons in cylindrical quantum dots of semiconductors," Phys. Rev. B 47, 1383-1391 (1993).
[CrossRef]

1991 (1)

Y. Kayanuma, "Wannier excitons in low-dimensional microstructures: Shape dependence of the quantum size effect," Phys. Rev. B 44, 13085-13088 (1991).
[CrossRef]

1985 (1)

1971 (1)

P. Lawaetz, "Valence-band parameters in cubic semiconductors," Phys. Rev. B 4, 3460-3467 (1971).
[CrossRef]

Alferov, Z. I.

M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop’ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment," Appl. Phys. Lett. 68, 979-981 (1996).
[CrossRef]

Alivisatos, A. P.

Y. Yin and A. P. Alivisatos, "Colloidal nanocrystal synthesis and the organic-inorganic interface," Nature 437, 664-670 (2005).
[CrossRef] [PubMed]

A. G. Kanaras, C. Sonnichsen, H. T. Liu, and A. P. Alivisatos, "Controlled synthesis of hyperbranched inorganic nanocrystals with rich three-dimensional structures," Nano Lett. 5, 2164-2167 (2005).
[CrossRef] [PubMed]

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L. W. Wang, and A. P. Alivisatos, "Colloidal nanocrystal heterostructures with linear and branched topology," Nature 430, 190-195 (2004).
[CrossRef] [PubMed]

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, "Controlled growth of tetrapod-branched inorganic nanocrystals," Nat. Mater. 2, 382-385 (2003).
[CrossRef] [PubMed]

L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, "Band gap variation of size- and shape-controlled colloidal CdSe quantum rods," Nano Lett. 1, 349-351 (2001).
[CrossRef]

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Baranov, A. V.

A. V. Fedorov, A. V. Baranov, and K. Inoue, "Two-photon transitions in systems with semiconductor quantum dots," Phys. Rev. B 54, 8627-8632 (1996).
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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
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C. Gan, M. Xiao, D. Battaglia, N. Pradhan, and X. G Peng, "Size dependence of nonlinear optical absorption and refraction of Mn-doped ZnSe nanocrystals," Appl. Phys. Lett. 91, 201103-201105 (2007).
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D. J. Norris, A. L. Efros, M. Rosen, and M. G. Bawendi, "Size dependence of exciton fine structure in CdSe quantum dots," Phys. Rev. B 53, 16347-16354 (1996).
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X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum qots for live cells, in vivo imaging, and diagnostics," Science 307, 538-544 (2005).
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M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop’ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment," Appl. Phys. Lett. 68, 979-981 (1996).
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M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, "Size-dependent two-photon excitation spectroscopy of CdSe nanocrystals," Phys. Rev. B 53, 12629-12632 (1996).
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B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, "In vivo imaging of quantum dots encapsulated in phospholipid micelles," Science 298, 1759-1762 (2002).
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D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, "Water-soluble quantum dots for multiphoton fluorescence imaging in vivo," Science 300, 1434-1436 (2003).
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F. Buonocore, D. Ninno, and G. Iadonisi, "Localized states in arbitrarily shaped quantum wire: a variation-perturbation technique," Phys. Stat. Sol.(b) 225, 343-352 (2001).
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D. Ninno, G. Iadonisi, and F. Buonocore, "Carrier localization and photoluminescence in porous silicon" Solid State Commun. 112, 521-525 (1999).
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M. B. Mohamed, C. Burda, and M. A. El-Sayed, "Shape dependent ultrafast relaxation dynamics of CdSe nanocrystals: Nanorods vs Nanodots," Nano Lett. 1, 589-593 (2001).
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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
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G. Cantele, D. Ninno, and G. Iadonisi, "Calculation of the infrared optical transitions in semiconductor ellipsoidal quantum dots," Nano Lett. 1, 121-124 (2001).
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G. Cantele, D. Ninno, and G. Iadonisi, "Confined states in ellipsoidal quantum dots," J. Phys.: Condens. Matter 12, 9019-9036 (2000).
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J. X. Cao, X. G. Gong, J. X. Zhong, and R. Q. Wu, "Sharp corners in the cross section of ultrathin Si nanowires," Phys. Rev. Lett. 97, 136105-136108 (2006).
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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
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X. Li, J. Embden, W. M. Chon, and M. Gu, "Enhanced two-photon absorption of CdS nanocrystal rods," Appl. Phys.Lett. 94, 103117-103119 (2009).
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D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, "Water-soluble quantum dots for multiphoton fluorescence imaging in vivo," Science 300, 1434-1436 (2003).
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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
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D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L. W. Wang, and A. P. Alivisatos, "Colloidal nanocrystal heterostructures with linear and branched topology," Nature 430, 190-195 (2004).
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D. Tari, M. De Giorgi, F. Della Sala, L. Carbone, R. Krahne, L. Manna, R. Cingolani, S. Kudera, and W. J. Parak, "Optical properties of tetrapod-shaped CdTe nanocrystals," Appl. Phys. Lett. 87, 224101-224103 (2005).
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L. Y. Pan, N. Tamai, K. Kamada, and S. Deki, "Nonlinear optical properties of thiol-capped CdTe quantum dots in nonresonant region," Appl. Phys. Lett. 91, 051902-051904 (2007).
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D. Tari, M. De Giorgi, F. Della Sala, L. Carbone, R. Krahne, L. Manna, R. Cingolani, S. Kudera, and W. J. Parak, "Optical properties of tetrapod-shaped CdTe nanocrystals," Appl. Phys. Lett. 87, 224101-224103 (2005).
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E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin and J. T. Devreese, "Development of an eight-band theory for quantum dot heterostructures," Phys. Rev. B 64, 245328-245343 (2001).
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V. V. Nikesh, A. Dharmadhikari, H. Ono, S. Nozaki, G. S. Kumar, and S. Mahamuni, "Optical nonlinearity of monodispersed, capped ZnS quantum particles," Appl. Phys. Lett. 84, 46024604-46024606 (2004).
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X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum qots for live cells, in vivo imaging, and diagnostics," Science 307, 538-544 (2005).
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B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou, and A. Libchaber, "In vivo imaging of quantum dots encapsulated in phospholipid micelles," Science 298, 1759-1762 (2002).
[CrossRef] [PubMed]

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D. J. Norris, A. L. Efros, M. Rosen, and M. G. Bawendi, "Size dependence of exciton fine structure in CdSe quantum dots," Phys. Rev. B 53, 16347-16354 (1996).
[CrossRef]

A. L. Efros and A. V. Arondina, "Band-edge absorption and luminescence of nonspherical nanometer-size crystals," Phys. Rev. B 47, 10005-10007 (1993).
[CrossRef]

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M. B. Mohamed, C. Burda, and M. A. El-Sayed, "Shape dependent ultrafast relaxation dynamics of CdSe nanocrystals: Nanorods vs Nanodots," Nano Lett. 1, 589-593 (2001).
[CrossRef]

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X. Li, J. Embden, W. M. Chon, and M. Gu, "Enhanced two-photon absorption of CdS nanocrystal rods," Appl. Phys.Lett. 94, 103117-103119 (2009).
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A. V. Fedorov, A. V. Baranov, and K. Inoue, "Two-photon transitions in systems with semiconductor quantum dots," Phys. Rev. B 54, 8627-8632 (1996).
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E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin and J. T. Devreese, "Development of an eight-band theory for quantum dot heterostructures," Phys. Rev. B 64, 245328-245343 (2001).
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E. P. Pokatilov, V. A. Fonoberov, V. M. Fomin and J. T. Devreese, "Development of an eight-band theory for quantum dot heterostructures," Phys. Rev. B 64, 245328-245343 (2001).
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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
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X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum qots for live cells, in vivo imaging, and diagnostics," Science 307, 538-544 (2005).
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C. Gan, M. Xiao, D. Battaglia, N. Pradhan, and X. G Peng, "Size dependence of nonlinear optical absorption and refraction of Mn-doped ZnSe nanocrystals," Appl. Phys. Lett. 91, 201103-201105 (2007).
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J. X. Cao, X. G. Gong, J. X. Zhong, and R. Q. Wu, "Sharp corners in the cross section of ultrathin Si nanowires," Phys. Rev. Lett. 97, 136105-136108 (2006).
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J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, "Coherent manipulation of coupled electron spins in semiconductor quantum dots," Science 309, 2180-2184 (2005).
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M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop’ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment," Appl. Phys. Lett. 68, 979-981 (1996).
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X. Li, J. Embden, W. M. Chon, and M. Gu, "Enhanced two-photon absorption of CdS nanocrystal rods," Appl. Phys.Lett. 94, 103117-103119 (2009).
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Guyot-Sionnest, P.

M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, "Size-dependent two-photon excitation spectroscopy of CdSe nanocrystals," Phys. Rev. B 53, 12629-12632 (1996).
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L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, "Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots," Phys. Rev. B 75, 075325-075332 (2007).
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C. Zhang, F. Zhang, S. Qian, N. Kumar, J. Hahm, and J. Xu, "Multiphoton absorption induced amplified spontaneous emission from biocatalyst-synthesized ZnO nanorods," Appl. Phys. Lett. 92, 233116-233118 (2008).
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J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, "Coherent manipulation of coupled electron spins in semiconductor quantum dots," Science 309, 2180-2184 (2005).
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Herz, E.

N. L. Thomas, E. Herz, O. Schöps and U. Woggon, "Exciton Fine Structure in Single CdSe Nanorods," Phys. Rev. Lett. 94, 016803-016806 (2005).
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M. E. Schmidt, S. A. Blanton, M. A. Hines, and P. Guyot-Sionnest, "Size-dependent two-photon excitation spectroscopy of CdSe nanocrystals," Phys. Rev. B 53, 12629-12632 (1996).
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L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, "Band gap variation of size- and shape-controlled colloidal CdSe quantum rods," Nano Lett. 1, 349-351 (2001).
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Hughes, S. M.

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L. W. Wang, and A. P. Alivisatos, "Colloidal nanocrystal heterostructures with linear and branched topology," Nature 430, 190-195 (2004).
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G. Cantele, D. Ninno, and G. Iadonisi, "Calculation of the infrared optical transitions in semiconductor ellipsoidal quantum dots," Nano Lett. 1, 121-124 (2001).
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G. Cantele, D. Ninno, and G. Iadonisi, "Confined states in ellipsoidal quantum dots," J. Phys.: Condens. Matter 12, 9019-9036 (2000).
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D. Ninno, G. Iadonisi, and F. Buonocore, "Carrier localization and photoluminescence in porous silicon" Solid State Commun. 112, 521-525 (1999).
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A. V. Fedorov, A. V. Baranov, and K. Inoue, "Two-photon transitions in systems with semiconductor quantum dots," Phys. Rev. B 54, 8627-8632 (1996).
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G. C. Xing, W. Ji, Y. G. Zheng, and J. Y. Ying, "Two- and three-photon absorption of semiconductor quantum dots in the vicinity of half of lowest exciton energy," Appl. Phys. Lett. 93, 241114-241116 (2008).
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J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, "Coherent manipulation of coupled electron spins in semiconductor quantum dots," Science 309, 2180-2184 (2005).
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L. Y. Pan, N. Tamai, K. Kamada, and S. Deki, "Nonlinear optical properties of thiol-capped CdTe quantum dots in nonresonant region," Appl. Phys. Lett. 91, 051902-051904 (2007).
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A. G. Kanaras, C. Sonnichsen, H. T. Liu, and A. P. Alivisatos, "Controlled synthesis of hyperbranched inorganic nanocrystals with rich three-dimensional structures," Nano Lett. 5, 2164-2167 (2005).
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T. Yatsui, S. Sangu, K. Kobayashi, T. Kawazoe, M. Ohtsu, J. Yoo, and G. C. Yi, "Nanophotonic energy up conversion using ZnO nanorod double-quantum-well structures," Appl. Phys. Lett. 94, 083113-083115 (2009).
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J. Kim, P. S. Nair, C. Y. Wong, and G. D. Scholes, "Sizing up the exciton in complex-shaped semiconductor nanocrystals," Nano Lett. 7, 3884-3890 (2007).
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T. Yatsui, S. Sangu, K. Kobayashi, T. Kawazoe, M. Ohtsu, J. Yoo, and G. C. Yi, "Nanophotonic energy up conversion using ZnO nanorod double-quantum-well structures," Appl. Phys. Lett. 94, 083113-083115 (2009).
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M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop’ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment," Appl. Phys. Lett. 68, 979-981 (1996).
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D. Tari, M. De Giorgi, F. Della Sala, L. Carbone, R. Krahne, L. Manna, R. Cingolani, S. Kudera, and W. J. Parak, "Optical properties of tetrapod-shaped CdTe nanocrystals," Appl. Phys. Lett. 87, 224101-224103 (2005).
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D. Tari, M. De Giorgi, F. Della Sala, L. Carbone, R. Krahne, L. Manna, R. Cingolani, S. Kudera, and W. J. Parak, "Optical properties of tetrapod-shaped CdTe nanocrystals," Appl. Phys. Lett. 87, 224101-224103 (2005).
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Kumar, G. S.

V. V. Nikesh, A. Dharmadhikari, H. Ono, S. Nozaki, G. S. Kumar, and S. Mahamuni, "Optical nonlinearity of monodispersed, capped ZnS quantum particles," Appl. Phys. Lett. 84, 46024604-46024606 (2004).
[CrossRef]

Kumar, N.

C. Zhang, F. Zhang, S. Qian, N. Kumar, J. Hahm, and J. Xu, "Multiphoton absorption induced amplified spontaneous emission from biocatalyst-synthesized ZnO nanorods," Appl. Phys. Lett. 92, 233116-233118 (2008).
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Kumar, S.

S. Kumar and T. Nann, "Shape control of II-VI semiconductor nanomaterials," Small 2, 316-329 (2006).
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J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, "Coherent manipulation of coupled electron spins in semiconductor quantum dots," Science 309, 2180-2184 (2005).
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D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, "Water-soluble quantum dots for multiphoton fluorescence imaging in vivo," Science 300, 1434-1436 (2003).
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M. Grundmann, N. N. Ledentsov, O. Stier, D. Bimberg, V. M. Ustinov, P. S. Kop’ev, and Z. I. Alferov, "Excited states in self-organized InAs/GaAs quantum dots: Theory and experiment," Appl. Phys. Lett. 68, 979-981 (1996).
[CrossRef]

Lee, S. M.

S. M. Lee, S. N. Cho, and J. Cheon, "Anisotropic shape control of colloidal inorganic nanocrystals," Adv. Mater. 15, 441-444 (2003).
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D. L. Yao, G. Zhang, and B. W. Li, "A universal expression of band gap for silicon nanowires of different cross-section geometries," Nano Lett. 8, 4557-4561 (2008).
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D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L. W. Wang, and A. P. Alivisatos, "Colloidal nanocrystal heterostructures with linear and branched topology," Nature 430, 190-195 (2004).
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J. Li and L. W. Wang, "Shape effects on electronic states of nanocrystals," Nano Lett. 3, 1357-1363 (2003).
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Li, J. J.

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, "Quantum qots for live cells, in vivo imaging, and diagnostics," Science 307, 538-544 (2005).
[CrossRef] [PubMed]

Li, L. S.

L. S. Li, J. T. Hu, W. D. Yang, and A. P. Alivisatos, "Band gap variation of size- and shape-controlled colloidal CdSe quantum rods," Nano Lett. 1, 349-351 (2001).
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Figures (6)

Fig. 1.
Fig. 1.

Schematic diagrams of semiconductor NCs with four shapes.

Fig. 2.
Fig. 2.

TPA cross-sections of cylindrical and spherical CdS NCs plotted as a function of excitation wavelength. The blue circles and red squares are the experimental data in Ref. [32].

Fig. 3.
Fig. 3.

Comparisons of TPA spectra between CdSe NCs and ZnO NCs with four shapes: (a) sphere vs. cylinder; (b) cube vs. cuboid; and (c) cube vs. sphere.

Fig. 4.
Fig. 4.

(a) Lowest transition energy versus diameter (or width); and (b) TPA cross-section contributed by the lowest transition versus diameter (or width). The aspect ratios for cylinder and cuboid are V=3.

Fig. 5.
Fig. 5.

Size dependence of TPA cross-section for (a) CdSe NCs at 780 nm and (b) ZnO NCs at 532 nm.

Fig. 6.
Fig. 6.

TPA cross-section vs. width (or diameter) and length at 780 nm for CdSe NCs and at 532 nm for ZnO NCs with the shapes of (a) cylinder and (b) cuboid.

Tables (1)

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Table 1. Parameters used in the Calculations for the Investigated Materials

Equations (12)

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W(2)=2πΣv1,v0Mv1,v02δ(Ev1Ev02ω),
Mv1,v0=Σv2Hv1,v2intHv2,v0intEv2Ev0ωiγv2.
a,nlmHinta,nlm=ieAmac×B,
B={2Dμnlμnl(μnl)2(μnl)2[δm,m(δn,n+1+δn,n1)ex+iδm,m(δn,n1δn,n+1)ey]
+1Lmδn,nδl,l(1δm,m)[1cos(m+m)πm+m+1cos(mm)πmm]ez},
B=[1Dnδm,mδl,l(1δn,n)(1cos(n+n)n+n+1cos(nn)πn'n)ex
+1Dlδm,mδn,n(1δl,l)(1cos(l+l)πl+l+1cos(ll)πll)ey
+ 1L m δn,n δl,l ( 1δm,m ) (1cos(m+m)πm+m+1cos(mm)πmm) ez ] .
α2=4 ω NI2 Wˉ(2) f (a)da,
α2=8πωNεω(2πe2Pcω2)2Σj=13Fc,hj,
Fc,hj=Σnlm,nlmB2Tf(a)δ(Ev0hjEv1c2ω)da,
T=1mc1(Ev0cEv0hjωiγv0)+1mhj1(Ev0hjEv1hj+ω+iγv1),

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