Abstract

Excitons of CdTe tetrapod-shaped nanocrystals are theoretically analyzed. Individual electron and hole states are calculated by solving one-particle Schrödinger equation by the finite element method with the single-band effective-mass approximation and exciton states are obtained by numerical diagonalization of the configuration interaction Hamiltonian. Spatial symmetries of the exciton states are related to those of the one-particle states by group theory and verified by numerical calculation. It is shown that the lowest exciton state is an optically active A 1 exciton. Optical absorption spectra are calculated and compared with available experimental data.

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  1. L. Manna, E. C. Scher, and A. P. Alivisatos, “Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals,” J. Am. Chem. Soc. 122, 12700–12706 (2000).
    [Crossref]
  2. A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
    [Crossref] [PubMed]
  3. L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, “Controlled growth of tetrapod-branched inorganic nanocrystals,” Nature Mat. 2, 382–385 (2003).
    [Crossref]
  4. M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
    [Crossref]
  5. D. Tarì, 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, 224101/1–224101/3 (2005).
    [Crossref]
  6. S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
    [Crossref] [PubMed]
  7. D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
    [Crossref]
  8. G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
    [Crossref]
  9. M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
    [Crossref] [PubMed]
  10. R. B. Vasiliev, D. N. Dirin, and A. M. Gaskov, “Temperature effect on the growth of colloidal CdTe nanotetrapods,” Mendeleev Commun. 19, 126–127 (2009).
    [Crossref]
  11. P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
    [Crossref] [PubMed]
  12. D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
    [Crossref] [PubMed]
  13. C. L. Choi, K. J. Koski, S. Sivasankar, and A. P. Alivisatos, “Strain-dependent photoluminescence behavior of CdSe/CdS nanocrystals with spherical, linear, and branched topologies,” Nano Lett. 9, 3544–3549 (2009).
    [Crossref] [PubMed]
  14. A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
    [Crossref]
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    [Crossref]
  16. Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
    [Crossref]
  17. Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
    [Crossref]
  18. J.-B. Li and L.-W. Wang, “Shape effects on electronic states of nanocrystals,” Nano Lett. 3, 1357–1363 (2003).
    [Crossref]
  19. D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L.-W. Wang, and P. Alivisatos, “Colloidal nanocrystal heterostructures with linear and branched topology,” Nature 430, 190–195 (2004).
    [Crossref] [PubMed]
  20. A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
    [Crossref] [PubMed]
  21. J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
    [Crossref] [PubMed]
  22. C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
    [Crossref]
  23. F. Bechstedt and R. Enderlein, Semiconductor Surfaces and Interfaces: Their Atomic and Electronic Structures (Akademie-Verlag, Berlin, 1988).
  24. S.-H. Wei and S. B. Zhang, “Structure stability and carrier localization in CdX (X=S, Se, Te) semiconductors”, Phys. Rev. B 62, 6944–6947 (2000).
    [Crossref]
  25. S. AdachiProperties of Group-IV, III–V and II–VI Semiconductors (Wiley, Chichester, 2005) P.218.
  26. T. Inui, Y. Tanabe, and Y. Onodera, Group theory and Its Applications in Physics (Springer-Verlag, Berlin1990).
  27. Y. Yao, T. Ochiai, T. Mano, T. Kuroda, T. Noda, N. Koguchi, and K. Sakoda, “Electronic structure of GaAs/AlGaAs quantum double rings in lateral electric field,” Chin. Opt. Lett. 7, 882–885 (2009).
    [Crossref]

2010 (2)

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

2009 (8)

Y. Yao, T. Ochiai, T. Mano, T. Kuroda, T. Noda, N. Koguchi, and K. Sakoda, “Electronic structure of GaAs/AlGaAs quantum double rings in lateral electric field,” Chin. Opt. Lett. 7, 882–885 (2009).
[Crossref]

R. B. Vasiliev, D. N. Dirin, and A. M. Gaskov, “Temperature effect on the growth of colloidal CdTe nanotetrapods,” Mendeleev Commun. 19, 126–127 (2009).
[Crossref]

C. L. Choi, K. J. Koski, S. Sivasankar, and A. P. Alivisatos, “Strain-dependent photoluminescence behavior of CdSe/CdS nanocrystals with spherical, linear, and branched topologies,” Nano Lett. 9, 3544–3549 (2009).
[Crossref] [PubMed]

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
[Crossref]

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

2008 (2)

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

2007 (1)

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

2006 (2)

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

2005 (4)

M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

2004 (1)

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

2003 (2)

J.-B. Li and L.-W. Wang, “Shape effects on electronic states of nanocrystals,” Nano Lett. 3, 1357–1363 (2003).
[Crossref]

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, “Controlled growth of tetrapod-branched inorganic nanocrystals,” Nature Mat. 2, 382–385 (2003).
[Crossref]

2000 (2)

L. Manna, E. C. Scher, and A. P. Alivisatos, “Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals,” J. Am. Chem. Soc. 122, 12700–12706 (2000).
[Crossref]

S.-H. Wei and S. B. Zhang, “Structure stability and carrier localization in CdX (X=S, Se, Te) semiconductors”, Phys. Rev. B 62, 6944–6947 (2000).
[Crossref]

Adachi, S.

S. AdachiProperties of Group-IV, III–V and II–VI Semiconductors (Wiley, Chichester, 2005) P.218.

Alivisatos, A. P.

C. L. Choi, K. J. Koski, S. Sivasankar, and A. P. Alivisatos, “Strain-dependent photoluminescence behavior of CdSe/CdS nanocrystals with spherical, linear, and branched topologies,” Nano Lett. 9, 3544–3549 (2009).
[Crossref] [PubMed]

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, “Controlled growth of tetrapod-branched inorganic nanocrystals,” Nature Mat. 2, 382–385 (2003).
[Crossref]

L. Manna, E. C. Scher, and A. P. Alivisatos, “Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals,” J. Am. Chem. Soc. 122, 12700–12706 (2000).
[Crossref]

Alivisatos, P.

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

Aloni, S.

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

Ambrozevich, S. A.

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

Bechstedt, F.

F. Bechstedt and R. Enderlein, Semiconductor Surfaces and Interfaces: Their Atomic and Electronic Structures (Akademie-Verlag, Berlin, 1988).

Becker, K.

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

Biasiucci, M.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Braun, M.

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

Carbone, L.

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

Carlino, E.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Cheng, G.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Choi, C. L.

C. L. Choi, K. J. Koski, S. Sivasankar, and A. P. Alivisatos, “Strain-dependent photoluminescence behavior of CdSe/CdS nanocrystals with spherical, linear, and branched topologies,” Nano Lett. 9, 3544–3549 (2009).
[Crossref] [PubMed]

Cingolani, R.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

Cucolo, A. M.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Cui, Y.

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

Da Como, E.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

De Giorgi, M.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
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D. Tarì, 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, 224101/1–224101/3 (2005).
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DeRocher, K. A.

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
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R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
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A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
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C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
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J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
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Fiore, A.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
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A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
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Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
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Gaskov, A. M.

R. B. Vasiliev, D. N. Dirin, and A. M. Gaskov, “Temperature effect on the growth of colloidal CdTe nanotetrapods,” Mendeleev Commun. 19, 126–127 (2009).
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Gaskovb, A. M.

R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
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Giannini, C.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
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Gigli, G.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
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Goodman, M. D.

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

Huang, J.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
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Hughes, S. M.

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L.-W. Wang, and P. Alivisatos, “Colloidal nanocrystal heterostructures with linear and branched topology,” Nature 430, 190–195 (2004).
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Inui, T.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and Its Applications in Physics (Springer-Verlag, Berlin1990).

Johnson, J. C.

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

Khokhlov, E. M.

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

Koeppe, R.

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
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Koski, K. J.

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M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

Kudera, S.

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
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Kuroda, T.

Lagoudakis, P. G.

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Lanzani, G.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Li, J.

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L.-W. Wang, and P. Alivisatos, “Colloidal nanocrystal heterostructures with linear and branched topology,” Nature 430, 190–195 (2004).
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Li, K.

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

Li, Y.

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
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C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

Lin, Z.

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

Lupo, M. G.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Lupton, J. M.

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Lutich, A. A.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

Malkmus, S.

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

Mallapragada, S. K.

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

Manna, L.

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L.-W. Wang, and 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,” Nature Mat. 2, 382–385 (2003).
[Crossref]

L. Manna, E. C. Scher, and A. P. Alivisatos, “Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals,” J. Am. Chem. Soc. 122, 12700–12706 (2000).
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Mano, T.

Mastria, R.

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Mauser, C.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

Meisel, A.

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, “Controlled growth of tetrapod-branched inorganic nanocrystals,” Nature Mat. 2, 382–385 (2003).
[Crossref]

Milliron, D. J.

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L.-W. Wang, and 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,” Nature Mat. 2, 382–385 (2003).
[Crossref]

Morello, G.

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

Müller, J.

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Nelson, J. H.

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

Nobile, C.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Noda, T.

Ochiai, T.

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and Its Applications in Physics (Springer-Verlag, Berlin1990).

Parak, W. J.

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

Peng, P.

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

Pompa, P. P.

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

Rogach, A. L.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Sadtler, B.

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

Sakoda, K.

Saykally, R. J.

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

Scher, E. C.

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, “Controlled growth of tetrapod-branched inorganic nanocrystals,” Nature Mat. 2, 382–385 (2003).
[Crossref]

L. Manna, E. C. Scher, and A. P. Alivisatos, “Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals,” J. Am. Chem. Soc. 122, 12700–12706 (2000).
[Crossref]

Schindler, F.

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Shevchenko, E. V.

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

Shul’ga, A. S.

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

Sivasankar, S.

C. L. Choi, K. J. Koski, S. Sivasankar, and A. P. Alivisatos, “Strain-dependent photoluminescence behavior of CdSe/CdS nanocrystals with spherical, linear, and branched topologies,” Nano Lett. 9, 3544–3549 (2009).
[Crossref] [PubMed]

Sokolikova, M. S.

R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
[Crossref]

Talapin, D. V.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and Its Applications in Physics (Springer-Verlag, Berlin1990).

Tarì, D.

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

Vaneski, A.

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

Vasiliev, R. B.

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
[Crossref]

R. B. Vasiliev, D. N. Dirin, and A. M. Gaskov, “Temperature effect on the growth of colloidal CdTe nanotetrapods,” Mendeleev Commun. 19, 126–127 (2009).
[Crossref]

Vitukhnovsky, A. G.

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

Vitukhnovskyc, A. G.

R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
[Crossref]

Wang, J.

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

Wang, L.-W.

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

J.-B. Li and L.-W. Wang, “Shape effects on electronic states of nanocrystals,” Nano Lett. 3, 1357–1363 (2003).
[Crossref]

Wang, Y.

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

Wei, S.-H.

S.-H. Wei and S. B. Zhang, “Structure stability and carrier localization in CdX (X=S, Se, Te) semiconductors”, Phys. Rev. B 62, 6944–6947 (2000).
[Crossref]

Weller, H.

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Yao, Y.

Yin, L.

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Yudson, V. I.

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

Zhang, S. B.

S.-H. Wei and S. B. Zhang, “Structure stability and carrier localization in CdX (X=S, Se, Te) semiconductors”, Phys. Rev. B 62, 6944–6947 (2000).
[Crossref]

Zhao, L.

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

ACS Nano (1)

M. D. Goodman, L. Zhao, K. A. DeRocher, J. Wang, S. K. Mallapragada, and Z. Lin, “Self-assembly of CdTe tetrapods into network monolayers at the air/water interface,” ACS Nano 4, 2043–2050 (2010).
[Crossref] [PubMed]

Adv. Mater. (1)

Y. Li, R. Mastria, A. Fiore, C. Nobile, L. Yin, M. Biasiucci, G. Cheng, A. M. Cucolo, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of heterostructured tetrapod-shaped CdSe/CdTe nanocrystals using C60 interlayer,” Adv. Mater. 21, 4461–4466 (2009).
[Crossref]

Appl. Phys. Lett. (4)

Y. Li, R. Mastria, K. Li, A. Fiore, Y. Wang, R. Cingolani, L. Manna, and G. Gigli, “Improved photovoltaic performance of bilayer heterojunction photovoltaic cells by triplet materials and tetrapod-shaped colloidal nanocrystals doping,” Appl. Phys. Lett. 95, 043101/1–043101/3 (2009).
[Crossref]

D. Tarì, M. De Giorgi, P. P. Pompa, L. Carbone, L. Manna, S. Kudera, and R. Cingolani, “Exciton transitions in tetrapod-shaped CdTe nanocrystals investigated by photomodulated transmittance spectroscopy,” Appl. Phys. Lett. 89, 094104/1–094104/3 (2006).
[Crossref]

G. Morello, D. Tarì, L. Carbone, L. Manna, R. Cingolani, and M. De Giorgi, “Radiative recombination dynamics in tetrapod-shaped CdTe nanocrystals: Evidence for a photoinduced screening of the internal electric field,” Appl. Phys. Lett. 92, 191905/1–191905/3 (2008).
[Crossref]

D. Tarì, 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, 224101/1–224101/3 (2005).
[Crossref]

Chin. Opt. Lett. (1)

J. Am. Chem. Soc. (2)

L. Manna, E. C. Scher, and A. P. Alivisatos, “Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals,” J. Am. Chem. Soc. 122, 12700–12706 (2000).
[Crossref]

A. Fiore, R. Mastria, M. G. Lupo, G. Lanzani, C. Giannini, E. Carlino, G. Morello, M. De Giorgi, Y. Li, R. Cingolani, and L. Manna, “Tetrapod-shaped colloidal nanocrystals of II–VI semiconductors prepared by seeded growth,” J. Am. Chem. Soc. 131, 2274–2282 (2009).
[Crossref] [PubMed]

J. Phys. Chem. B (1)

S. Malkmus, S. Kudera, L. Manna, W. J. Parak, and M. Braun, “Electron-hole dynamics in CdTe tetrapods,” J. Phys. Chem. B 110, 17334–17338 (2006).
[Crossref] [PubMed]

Mendeleev Commun. (2)

R. B. Vasiliev, D. N. Dirin, and A. M. Gaskov, “Temperature effect on the growth of colloidal CdTe nanotetrapods,” Mendeleev Commun. 19, 126–127 (2009).
[Crossref]

R. B. Vasiliev, D. N. Dirin, M. S. Sokolikova, S. G. Dorofeev, A. G. Vitukhnovskyc, and A. M. Gaskovb, “Growth of near-IR luminescent colloidal CdTe/CdS nanoheterostructures based on CdTe tetrapods,” Mendeleev Commun. 19, 128–130 (2009).
[Crossref]

Microelectron. J. (1)

M. De Giorgi, D. Tarì, L. Manna, R. Krahne, and R. Cingolani, “Optical properties of colloidal nanocrystal spheres and tetrapods,” Microelectron. J. 36552–554 (2005).
[Crossref]

Nano Lett. (5)

J.-B. Li and L.-W. Wang, “Shape effects on electronic states of nanocrystals,” Nano Lett. 3, 1357–1363 (2003).
[Crossref]

P. Peng, D. J. Milliron, S. M. Hughes, J. C. Johnson, A. P. Alivisatos, and R. J. Saykally, “Femtosecond spectroscopy of carrier relaxation dynamics in type II CdSe/CdTe tetrapod heteronanostructures,” Nano Lett. 5, 1809–1813 (2005).
[Crossref] [PubMed]

D. V. Talapin, J. H. Nelson, E. V. Shevchenko, S. Aloni, B. Sadtler, and A. P. Alivisatos, “Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies,” Nano Lett. 7, 2951–2959 (2007).
[Crossref] [PubMed]

C. L. Choi, K. J. Koski, S. Sivasankar, and A. P. Alivisatos, “Strain-dependent photoluminescence behavior of CdSe/CdS nanocrystals with spherical, linear, and branched topologies,” Nano Lett. 9, 3544–3549 (2009).
[Crossref] [PubMed]

J. Müller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, J. Feldmann, D. V. Talapin, and H. Weller, “Wave function engineering in elongated semiconductor nanocrystals with heterogeneous carrier confinement,” Nano Lett. 52044–2049 (2005).
[Crossref] [PubMed]

Nano. Lett. (1)

A. A. Lutich, C. Mauser, E. Da Como, J. Huang, A. Vaneski, D. V. Talapin, A. L. Rogach, and J. Feldmann, “Multiexcitonic dual emission in CdSe/CdS tetrapods and nanorods,” Nano. Lett. 104646–4650 (2010).
[Crossref] [PubMed]

Nature (1)

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

Nature Mat. (1)

L. Manna, D. J. Milliron, A. Meisel, E. C. Scher, and A. P. Alivisatos, “Controlled growth of tetrapod-branched inorganic nanocrystals,” Nature Mat. 2, 382–385 (2003).
[Crossref]

Phys. Lett. A (1)

A. G. Vitukhnovsky, A. S. Shul’ga, S. A. Ambrozevich, E. M. Khokhlov, R. B. Vasiliev, D. N. Dirin, and V. I. Yudson, “Effect of branching of tetrapod-shaped CdTe/CdSe nanocrystal heterostructures on their luminescence,” Phys. Lett. A 373, 2287–2290 (2009).
[Crossref]

Phys. Rev. B (2)

C. Mauser, T. Limmer, E. Da Como, K. Becker, A. L. Rogach, J. Feldmann, and D. V. Talapin, “Anisotropic optical emission of single CdSe/CdS tetrapod heterostructures: Evidence for a wavefunction symmetry breaking,” Phys. Rev. B 77, 153303 (2008).
[Crossref]

S.-H. Wei and S. B. Zhang, “Structure stability and carrier localization in CdX (X=S, Se, Te) semiconductors”, Phys. Rev. B 62, 6944–6947 (2000).
[Crossref]

Other (3)

S. AdachiProperties of Group-IV, III–V and II–VI Semiconductors (Wiley, Chichester, 2005) P.218.

T. Inui, Y. Tanabe, and Y. Onodera, Group theory and Its Applications in Physics (Springer-Verlag, Berlin1990).

F. Bechstedt and R. Enderlein, Semiconductor Surfaces and Interfaces: Their Atomic and Electronic Structures (Akademie-Verlag, Berlin, 1988).

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Figures (7)

Fig. 1
Fig. 1

(a) Structure of the CdTe tetrapod that consists of four cylindrical arms, which has the wurzite crystal structure, and spherical central region, which has the zinc blende structure. The diameter and length of the arms are denoted by D and L. The diameter of the central region is assumed to be the same as the arms. Three values of D (1.9, 2.2, and 2.5 nm) and L (8.3, 9.0, and 9.7 nm) are used in the numerical calculation. (b) Confinement potential of electron and hole. We assume that the electrons are confined by a potential barrier whose height is equal to the electron affinity of CdTe whereas the holes are confined by an infinite potential barrier [5]. −4.18 eV is assumed for the ZB CdTe electron affinity, which was derived from the XPS experiment [23], whereas 1.5 eV is assumed for the ZB CdTe bandgap [5]. As for the band offset, we use 65 meV for the conduction band and 18 meV for the valence band, which were obtained by theoretical calculation [24]. For effective masses of electron and heavy hole, we assumed m e *   =   0.11   ×   m 0 and m h *   =   0.69   ×   m 0 , where m 0 is the genuin electron mass. [25]

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Fig. 2
Fig. 2

Envelope functions of (a) the lowest and (b) the second lowest A 1 states of electron.

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Fig. 3
Fig. 3

Convergence of the lowest 20 energy eigenvalues of the spin-triplet excitons. (L = 9.0 nm, D = 2.2 nm)

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Fig. 4
Fig. 4

(a) D and (b) L dependence of the spin-singlet exciton energy. The lowest twenty exciton states are shown, whose symmetries are A 1 or T 2.

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Fig. 5
Fig. 5

(a) D and (b) L dependence of the spin-triplet exciton energy. The lowest twenty exciton states are shown, whose symmetries are A 1 or T 2.

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Fig. 6
Fig. 6

Absorption spectra of CdTe tetrapods with different D values, where the lowest eight spin-singlet A 1 excitons are taken into consideration. A spectral width of 60 meV (FWHM) due to size distribution is rather arbitrarily assumed for each exciton transition.

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Fig. 7
Fig. 7

Absorption spectra of CdTe tetrapods with different L values, where the lowest eight spin-singlet A 1 excitons are taken into consideration. A spectral width of 60 meV (FWHM) due to size distribution is rather arbitrarily assumed for each exciton transition.

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Tables (4)

Tables Icon

Table 1 Character table of point group Td . The standard notations for symmetry operations are used: E for the identity operation, Cn for rotation by 360/n degrees, I for inversion, and σd for mirror reflection about a diagonal plane. The numbers in front of the symbols of the symmetry operations denote the number of conjugate operations.

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Table 2 Symmetry of the electron-hole pair state. The left column shows the individual symmetry of electron and hole envelope functions. The middle column is the character of the pair states. The right column shows the symmetry of the pair states obtained by the reduction procedure.

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Tables Icon

Table 3 The lowest twenty energy levels of a confined electron and hole in the tetrapod with L = 9.0 nm and D = 2.2 nm. The origin of the energy eigenvalues of the electron (hole) is the bottom of the conduction band (top of the valence band) in the core of the tetrapod with the zinc blende crystal structure.

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Table 4 The lowest 44 electron-hole pair states for L = 9.0 nm and D = 2.2 nm without considering the Coulomb potential energy. Only A 1 states have non-zero transition dipole moments and contribute to the optical transition.

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Equations (18)

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

ψ e   ( r e )   =   φ e ( r e ) u e ( r e ) ,
ψ h   ( r h )   =   φ h ( r h ) u h ( r h ) ,
ℋ e φ e ( r e )   ≡   { − h ¯ 2 Δ e 2 m e *   +   V e ( r e ) }   φ e ( r e )   =   E e φ e ( r e ) ,
ℋ h φ h ( r h )   ≡   { − h ¯ 2 Δ h 2 m h *   +   V h ( r h ) }   φ h ( r h )   =   E h φ h ( r h ) ,
R ℋ e , h R − 1   =   ℋ e , h   ( ∀   R   ∈   T d ) .
ℋ X   Ψ   ( r e ,   r h )   ≡   ( ℋ e   +   ℋ h   −   e 0 2 4 π ɛ 0 ɛ | r e   −   r h | )   Ψ ( r e ,   r h )   =   E X Ψ ( r e , r h ) ,
Ψ ( r e ,   r h )   =   ∑ i , j a ij   φ e ( i )   ( r e ) φ h ( j )   ( r h ) ,
R ℋ X R − 1   =   ℋ X   ( ∀ R   ∈   T d ) .
I o   =   ∫ d r φ e * ( r ) φ h ( r ) .
| ij ( s ) 〉   =   1 2   { | φ e ( i )   ↑   φ h ( j )   ↓ 〉   −   | ϕ e ( i )   ↓   φ h ( j )   ↑ 〉 } ,
〈 kl ( s ) | ℋ 2 | ij ( s ) 〉   =   〈 kj | ℋ 2 | il 〉   −   2 〈 jk | ℋ 2 | il 〉 ,
〈 kj | ℋ 2 | il 〉   =   − ∬ d r 1 d r 2 φ h ( j ) *   ( r 2 ) φ e ( k ) *   ( r 1 )   e 0 2 ɛ 0 ɛ | r 1   −   r 2 |   φ e ( i )   ( r 1 )   φ h ( l )   ( r 2 ) ,
| i j ( t ) 〉 = { | φ e ( i ) ↑ φ h ( j ) ↑ 〉 , 1 2 { | φ e ( i ) ↑ φ h ( j ) ↓ 〉 + | φ e ( i ) ↓ φ h ( j ) ↑ 〉 } , | φ e ( i ) ↓ φ h ( j ) ↓ 〉 ,
〈 kl ( t ) | ℋ 2 | i j ( t ) 〉   =   〈 k j | ℋ 2 | il 〉 .
ϕ A 1   =   1 2   ( ϕ 1   +   ϕ 2   +   ϕ 3   +   ϕ 4 ) .
ϕ T 2 ( 1 )   =   1 2   ( ϕ 1   +   ϕ 2   −   ϕ 3   −   ϕ 4 ) ,
ϕ T 2 ( 2 )   =   1 2   ( ϕ 1   −   ϕ 2   +   ϕ 3   −   ϕ 4 ) ,
ϕ T 2 ( 3 )   =   1 2   ( ϕ 1   −   ϕ 2   −   ϕ 3   +   ϕ 4 ) .

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