Abstract

The electronic structure of spherical PbS and PbSe quantum dots is calculated with a four-band envelope-function formalism. This calculation accounts for both exciton energies and wave functions with the correct symmetry of the materials. The selection rules and the strength of the dipole transitions of lead-salt quantum dots are derived accounting for the symmetry of the band-edge Bloch functions of the lead salts. The calculated energies of the optically allowed exciton states are found to be in good agreement with experimental data. The effects of many-body perturbations, such as Coulomb interactions and intervalley scattering, are also discussed.

© 1997 Optical Society of America

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    [CrossRef]
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    [CrossRef]

1996 (3)

T. D. Krauss, F. W. Wise, and D. B. Tanner “Observation of coupled vibrational modes of a semiconductor nanocrystal,” Phys. Rev. Lett. 76, 1376–1379 (1996).
[CrossRef] [PubMed]

R. S. Kane, R. E. Cohen, and R. Silbey, “Theoretical study of the electronic structure of PbS nanoclusters,” J. Phys. Chem. 100, 7928–7932 (1996).
[CrossRef]

M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A. I. Ekimov, “Enhancement of electron–hole exchange interaction in CdSe nanocrystals: a quantum confinement effect,” Phys. Rev. B 53, 1336–1342 (1996).
[CrossRef]

1995 (3)

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[CrossRef]

S. Gorer, A. Albu-Yaron, and G. Hodes, “Quantum size effects in chemically deposited, nanocrystalline lead selenide films,” J. Phys. Chem. 99, 16442–16448 (1995).
[CrossRef]

1994 (3)

N. F. Borrelli and D. W. Smith, “Quantum confinement of PbS microcrystals in glass,” J. Non-Cryst. Solids 180, 25–31 (1994).
[CrossRef]

E. Roca, C. Trallero-Giner, and M. Cardona, “Polar optical vibrational modes in quantum dots,” Phys. Rev. B 49, 13704–13711 (1994).
[CrossRef]

S. Nomura, Y. Segawa, and T. Kobayashi “Confined excitons in a semiconductor quantum dot in a magnetic field,” Phys. Rev. B 49, 13571–13582 (1994).
[CrossRef]

1993 (3)

T. Takagahara, “Effects of dielectric confinement and electron–hole exchange interaction on excitonic states in semiconductor quantum dots,” Phys. Rev. B 47, 4569–4584 (1993).
[CrossRef]

J. L. Machol, F. W. Wise, R. C. Patel, and D. B. Tanner “Vibronic quantum beats in PbS microcrystallites,” Phys. Rev. B 48, 2819–2822 (1993).
[CrossRef]

H. Weller, “Quantized semiconductor particles: a novel states of matter for material science,” Adv. Mater. 5, 88–95 (1993).
[CrossRef]

1992 (1)

S. Nomura and T. Kobayashi, “Exciton-LO-phonon couplings in spherical semiconductor microcrystallites,” Phys. Rev. B 45, 1305–1316 (1992).
[CrossRef]

1990 (1)

M. T. Nenadović, M. I. Čomor, V. Vasić, and O. I. Mićić, “Transient bleaching of small PbS colloids. Influence of surface properties,” J. Phys. Chem. 94, 6390–6396 (1990).
[CrossRef]

1989 (1)

S. Gallardo, M. Gutierrez, A. Henglein, and E. Janata, “Photochemistry and radiation chemistry of colloidal semiconductors.34. Properties of Q-PbS,” Ber. Bunsenges. Phys. Chem. 93, 1080–1090 (1989).
[CrossRef]

1988 (1)

L. C. Andreani, F. Bassani, and A. Quattropani, “Longitudinal-transverse splitting in Wannier excitons and polariton states,” Nuovo Cimento 10, 1473–1486 (1988).
[CrossRef]

1987 (1)

Y. Wang, A. Suna, W. Mahler, and R. Kasowski, “PbS in polymers. From molecules to bulk solids,” J. Chem. Phys. 87, 7315–7322 (1987).
[CrossRef]

1979 (1)

H. Preier, “Recent advances in lead-chalcogenide diode lasers,” Appl. Phys. 20, 189–206 (1979).
[CrossRef]

1973 (1)

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

1967 (1)

M. M. Elcombe, “The crystal dynamics of lead sulphide,” Proc. R. Soc. London Ser. A 300, 210–217 (1967).
[CrossRef]

1966 (1)

D. L. Mitchell and R. F. Wallis, “Theoretical energy-band parameters for the lead salts,” Phys. Rev. 151, 581–595 (1966).
[CrossRef]

Albu-Yaron, A.

S. Gorer, A. Albu-Yaron, and G. Hodes, “Quantum size effects in chemically deposited, nanocrystalline lead selenide films,” J. Phys. Chem. 99, 16442–16448 (1995).
[CrossRef]

Andreani, L. C.

L. C. Andreani, F. Bassani, and A. Quattropani, “Longitudinal-transverse splitting in Wannier excitons and polariton states,” Nuovo Cimento 10, 1473–1486 (1988).
[CrossRef]

Bassani, F.

L. C. Andreani, F. Bassani, and A. Quattropani, “Longitudinal-transverse splitting in Wannier excitons and polariton states,” Nuovo Cimento 10, 1473–1486 (1988).
[CrossRef]

Bawendi, M. G.

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

Borrelli, N. F.

N. F. Borrelli and D. W. Smith, “Quantum confinement of PbS microcrystals in glass,” J. Non-Cryst. Solids 180, 25–31 (1994).
[CrossRef]

Cardona, M.

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[CrossRef]

E. Roca, C. Trallero-Giner, and M. Cardona, “Polar optical vibrational modes in quantum dots,” Phys. Rev. B 49, 13704–13711 (1994).
[CrossRef]

Chamarro, M.

M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A. I. Ekimov, “Enhancement of electron–hole exchange interaction in CdSe nanocrystals: a quantum confinement effect,” Phys. Rev. B 53, 1336–1342 (1996).
[CrossRef]

Chamberlain, M. P.

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[CrossRef]

Cohen, M. L.

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

Cohen, R. E.

R. S. Kane, R. E. Cohen, and R. Silbey, “Theoretical study of the electronic structure of PbS nanoclusters,” J. Phys. Chem. 100, 7928–7932 (1996).
[CrossRef]

Comor, M. I.

M. T. Nenadović, M. I. Čomor, V. Vasić, and O. I. Mićić, “Transient bleaching of small PbS colloids. Influence of surface properties,” J. Phys. Chem. 94, 6390–6396 (1990).
[CrossRef]

Efros, Al. L.

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

Ekimov, A. I.

M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A. I. Ekimov, “Enhancement of electron–hole exchange interaction in CdSe nanocrystals: a quantum confinement effect,” Phys. Rev. B 53, 1336–1342 (1996).
[CrossRef]

Elcombe, M. M.

M. M. Elcombe, “The crystal dynamics of lead sulphide,” Proc. R. Soc. London Ser. A 300, 210–217 (1967).
[CrossRef]

Gallardo, S.

S. Gallardo, M. Gutierrez, A. Henglein, and E. Janata, “Photochemistry and radiation chemistry of colloidal semiconductors.34. Properties of Q-PbS,” Ber. Bunsenges. Phys. Chem. 93, 1080–1090 (1989).
[CrossRef]

Gorer, S.

S. Gorer, A. Albu-Yaron, and G. Hodes, “Quantum size effects in chemically deposited, nanocrystalline lead selenide films,” J. Phys. Chem. 99, 16442–16448 (1995).
[CrossRef]

Gourdon, C.

M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A. I. Ekimov, “Enhancement of electron–hole exchange interaction in CdSe nanocrystals: a quantum confinement effect,” Phys. Rev. B 53, 1336–1342 (1996).
[CrossRef]

Gutierrez, M.

S. Gallardo, M. Gutierrez, A. Henglein, and E. Janata, “Photochemistry and radiation chemistry of colloidal semiconductors.34. Properties of Q-PbS,” Ber. Bunsenges. Phys. Chem. 93, 1080–1090 (1989).
[CrossRef]

Henglein, A.

S. Gallardo, M. Gutierrez, A. Henglein, and E. Janata, “Photochemistry and radiation chemistry of colloidal semiconductors.34. Properties of Q-PbS,” Ber. Bunsenges. Phys. Chem. 93, 1080–1090 (1989).
[CrossRef]

Hodes, G.

S. Gorer, A. Albu-Yaron, and G. Hodes, “Quantum size effects in chemically deposited, nanocrystalline lead selenide films,” J. Phys. Chem. 99, 16442–16448 (1995).
[CrossRef]

Janata, E.

S. Gallardo, M. Gutierrez, A. Henglein, and E. Janata, “Photochemistry and radiation chemistry of colloidal semiconductors.34. Properties of Q-PbS,” Ber. Bunsenges. Phys. Chem. 93, 1080–1090 (1989).
[CrossRef]

Kane, R. S.

R. S. Kane, R. E. Cohen, and R. Silbey, “Theoretical study of the electronic structure of PbS nanoclusters,” J. Phys. Chem. 100, 7928–7932 (1996).
[CrossRef]

Kasowski, R.

Y. Wang, A. Suna, W. Mahler, and R. Kasowski, “PbS in polymers. From molecules to bulk solids,” J. Chem. Phys. 87, 7315–7322 (1987).
[CrossRef]

Kobayashi, T.

S. Nomura, Y. Segawa, and T. Kobayashi “Confined excitons in a semiconductor quantum dot in a magnetic field,” Phys. Rev. B 49, 13571–13582 (1994).
[CrossRef]

S. Nomura and T. Kobayashi, “Exciton-LO-phonon couplings in spherical semiconductor microcrystallites,” Phys. Rev. B 45, 1305–1316 (1992).
[CrossRef]

Kohn, S. E.

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

Krauss, T. D.

T. D. Krauss, F. W. Wise, and D. B. Tanner “Observation of coupled vibrational modes of a semiconductor nanocrystal,” Phys. Rev. Lett. 76, 1376–1379 (1996).
[CrossRef] [PubMed]

Kuno, M.

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

Lavallard, P.

M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A. I. Ekimov, “Enhancement of electron–hole exchange interaction in CdSe nanocrystals: a quantum confinement effect,” Phys. Rev. B 53, 1336–1342 (1996).
[CrossRef]

Lublinskaya, O.

M. Chamarro, C. Gourdon, P. Lavallard, O. Lublinskaya, and A. I. Ekimov, “Enhancement of electron–hole exchange interaction in CdSe nanocrystals: a quantum confinement effect,” Phys. Rev. B 53, 1336–1342 (1996).
[CrossRef]

Machol, J. L.

J. L. Machol, F. W. Wise, R. C. Patel, and D. B. Tanner “Vibronic quantum beats in PbS microcrystallites,” Phys. Rev. B 48, 2819–2822 (1993).
[CrossRef]

Mahler, W.

Y. Wang, A. Suna, W. Mahler, and R. Kasowski, “PbS in polymers. From molecules to bulk solids,” J. Chem. Phys. 87, 7315–7322 (1987).
[CrossRef]

Micic, O. I.

M. T. Nenadović, M. I. Čomor, V. Vasić, and O. I. Mićić, “Transient bleaching of small PbS colloids. Influence of surface properties,” J. Phys. Chem. 94, 6390–6396 (1990).
[CrossRef]

Mitchell, D. L.

D. L. Mitchell and R. F. Wallis, “Theoretical energy-band parameters for the lead salts,” Phys. Rev. 151, 581–595 (1966).
[CrossRef]

Nenadovic, M. T.

M. T. Nenadović, M. I. Čomor, V. Vasić, and O. I. Mićić, “Transient bleaching of small PbS colloids. Influence of surface properties,” J. Phys. Chem. 94, 6390–6396 (1990).
[CrossRef]

Nirmal, M.

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

Nomura, S.

S. Nomura, Y. Segawa, and T. Kobayashi “Confined excitons in a semiconductor quantum dot in a magnetic field,” Phys. Rev. B 49, 13571–13582 (1994).
[CrossRef]

S. Nomura and T. Kobayashi, “Exciton-LO-phonon couplings in spherical semiconductor microcrystallites,” Phys. Rev. B 45, 1305–1316 (1992).
[CrossRef]

Norris, D. J.

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

Patel, R. C.

J. L. Machol, F. W. Wise, R. C. Patel, and D. B. Tanner “Vibronic quantum beats in PbS microcrystallites,” Phys. Rev. B 48, 2819–2822 (1993).
[CrossRef]

Petroff, Y.

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

Preier, H.

H. Preier, “Recent advances in lead-chalcogenide diode lasers,” Appl. Phys. 20, 189–206 (1979).
[CrossRef]

Quattropani, A.

L. C. Andreani, F. Bassani, and A. Quattropani, “Longitudinal-transverse splitting in Wannier excitons and polariton states,” Nuovo Cimento 10, 1473–1486 (1988).
[CrossRef]

Roca, E.

E. Roca, C. Trallero-Giner, and M. Cardona, “Polar optical vibrational modes in quantum dots,” Phys. Rev. B 49, 13704–13711 (1994).
[CrossRef]

Rosen, M.

M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros, and M. Rosen “Observation of the “dark exciton” in CdSe quantum dots,” Phys. Rev. Lett. 75, 3728–3731 (1995).
[CrossRef] [PubMed]

Segawa, Y.

S. Nomura, Y. Segawa, and T. Kobayashi “Confined excitons in a semiconductor quantum dot in a magnetic field,” Phys. Rev. B 49, 13571–13582 (1994).
[CrossRef]

Shen, Y. R.

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

Silbey, R.

R. S. Kane, R. E. Cohen, and R. Silbey, “Theoretical study of the electronic structure of PbS nanoclusters,” J. Phys. Chem. 100, 7928–7932 (1996).
[CrossRef]

Smith, D. W.

N. F. Borrelli and D. W. Smith, “Quantum confinement of PbS microcrystals in glass,” J. Non-Cryst. Solids 180, 25–31 (1994).
[CrossRef]

Suna, A.

Y. Wang, A. Suna, W. Mahler, and R. Kasowski, “PbS in polymers. From molecules to bulk solids,” J. Chem. Phys. 87, 7315–7322 (1987).
[CrossRef]

Takagahara, T.

T. Takagahara, “Effects of dielectric confinement and electron–hole exchange interaction on excitonic states in semiconductor quantum dots,” Phys. Rev. B 47, 4569–4584 (1993).
[CrossRef]

Tanner, D. B.

T. D. Krauss, F. W. Wise, and D. B. Tanner “Observation of coupled vibrational modes of a semiconductor nanocrystal,” Phys. Rev. Lett. 76, 1376–1379 (1996).
[CrossRef] [PubMed]

J. L. Machol, F. W. Wise, R. C. Patel, and D. B. Tanner “Vibronic quantum beats in PbS microcrystallites,” Phys. Rev. B 48, 2819–2822 (1993).
[CrossRef]

Trallero-Giner, C.

M. P. Chamberlain, C. Trallero-Giner, and M. Cardona, “Theory of one-phonon Raman scattering in semiconductor microcrystallites,” Phys. Rev. B 51, 1680–1693 (1995).
[CrossRef]

E. Roca, C. Trallero-Giner, and M. Cardona, “Polar optical vibrational modes in quantum dots,” Phys. Rev. B 49, 13704–13711 (1994).
[CrossRef]

Tsang, Y.

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

Vasic, V.

M. T. Nenadović, M. I. Čomor, V. Vasić, and O. I. Mićić, “Transient bleaching of small PbS colloids. Influence of surface properties,” J. Phys. Chem. 94, 6390–6396 (1990).
[CrossRef]

Wallis, R. F.

D. L. Mitchell and R. F. Wallis, “Theoretical energy-band parameters for the lead salts,” Phys. Rev. 151, 581–595 (1966).
[CrossRef]

Wang, Y.

Y. Wang, A. Suna, W. Mahler, and R. Kasowski, “PbS in polymers. From molecules to bulk solids,” J. Chem. Phys. 87, 7315–7322 (1987).
[CrossRef]

Weller, H.

H. Weller, “Quantized semiconductor particles: a novel states of matter for material science,” Adv. Mater. 5, 88–95 (1993).
[CrossRef]

Wise, F. W.

T. D. Krauss, F. W. Wise, and D. B. Tanner “Observation of coupled vibrational modes of a semiconductor nanocrystal,” Phys. Rev. Lett. 76, 1376–1379 (1996).
[CrossRef] [PubMed]

J. L. Machol, F. W. Wise, R. C. Patel, and D. B. Tanner “Vibronic quantum beats in PbS microcrystallites,” Phys. Rev. B 48, 2819–2822 (1993).
[CrossRef]

Yu, P. Y.

S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen “Electronic band structure and optical properties of PbTe, PbSe, and PbS,” Phys. Rev. B 8, 1477–1488 (1973).
[CrossRef]

Adv. Mater. (1)

H. Weller, “Quantized semiconductor particles: a novel states of matter for material science,” Adv. Mater. 5, 88–95 (1993).
[CrossRef]

Appl. Phys. (1)

H. Preier, “Recent advances in lead-chalcogenide diode lasers,” Appl. Phys. 20, 189–206 (1979).
[CrossRef]

Ber. Bunsenges. Phys. Chem. (1)

S. Gallardo, M. Gutierrez, A. Henglein, and E. Janata, “Photochemistry and radiation chemistry of colloidal semiconductors.34. Properties of Q-PbS,” Ber. Bunsenges. Phys. Chem. 93, 1080–1090 (1989).
[CrossRef]

J. Chem. Phys. (1)

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Figures (13)

Fig. 1
Fig. 1

Band structure of bulk PbS (T=0 K) near the L point along the 〈111〉 direction, calculated by the empirical pseudopotential method (dashed curve), the four-band model (solid curve), and the parabolic effective-mass model (dotted curve). The lattice constant a is 5.93 Å. The bulk band gap of PbS at T =0 K is 0.28 eV.

Fig. 2
Fig. 2

Unperturbed energy levels of conduction and valence electrons of (a) PbS QD’s and (b) PbSe QD’s.

Fig. 3
Fig. 3

Plot of ϱ(k) for conduction electrons of PbS QD’s with j=1/2, π=1 (solid curve) and j=3/2, π=1 (dashed curve). The definition and explanation of ϱ(k) are given in the text.

Fig. 4
Fig. 4

Shifts of the energies of the conduction electrons with j =1/2, π=-1 (solid curve) and the valence electrons with j =1/2, π=1 (dashed curve) of (a) PbS QD’s and (b) PbSe QD’s owing to band anisotropy.

Fig. 5
Fig. 5

Shifts Δ|m|=1/2 (solid curve) and |Δ|m|=3/2| (dashed curve) of (a) the conduction-electron levels, |j=1/2,π=1 and |j =3/2,π=1, and (b) the valence-electron levels, |j=1/2,π =-1 and |j=3/2,π=-1, of PbS QD’s. See the text for the meaning of Δ|m|=1/2 and Δ|m|=3/2. The cusplike feature of the dashed curve in (b) results from the zero crossing of the shift Δ|m|=3/2.

Fig. 6
Fig. 6

Shifts Δ|m|=1/2 (solid curve) and |Δ|m|=3/2| (dashed curve) of (a) the conduction-electron levels, |j=1/2,π=1 and |j=3/2,π=1, and (b) the valence-electron levels, |j =1/2,π =-1 and |j=3/2,π=-1, of PbSe QD’s.

Fig. 7
Fig. 7

Comparison of the experimental values of the lowest exciton energy of PbS QD’s (triangles, solid circles, and squares) with the calculation by the four-band envelope-function formalism (solid curve), the hyperbolic-band model (dashed curve), the tight-binding calculation (crosses),and the parabolic effective-mass model (dotted curve).

Fig. 8
Fig. 8

Comparison of the energies of the first and second absorption peaks of PbS QD’s measured in this work (triangles) and Ref. 5 (squares) with the theoretical calculations by the four-band envelope-function formalism (solid curve), the hyperbolic-band model (dashed curve), and the parabolic effective-mass model (dotted curve). For the hyperbolic-band model, only the prediction of the lowest exciton energy can be given.

Fig. 9
Fig. 9

Absorption spectra and calculated transition strengths of PbS QD’s with diameters (a) 4.8 nm and (b) 7.6 nm.

Fig. 10
Fig. 10

Comparison of the experimental values of the lowest exciton energy of PbSe QD’s (bars) with the calculation by the four-band envelope-function formalism (solid curve), the hyperbolic-band model (dashed curve), and the parabolic effective-mass model (dotted curve).

Fig. 11
Fig. 11

(a) Coulomb energy shift in the lowest direct exciton states of PbS QD’s calculated with the four-band formalism. (b) Comparison of the experimental values of the first exciton energy (squares) with calculations that include (solid curve) and neglect (dotted curve) the energy of the Coulomb interaction.

Fig. 12
Fig. 12

Calculated energy of the triplet–singlet splitting in the lowest direct exciton states of PbS QD’s owing to the long-range exchange interaction.

Fig. 13
Fig. 13

Calculated energy splitting owing to the short-range exchange interaction in the lowest direct exciton states of PbS QD’s is plotted in units of the bulk exchange interaction strength constant J.

Tables (2)

Tables Icon

Table 1 Parameters of the k · p Hamiltonians [Eqs. (1) and (3)] of PbS and PbSea

Tables Icon

Table 2 Parameters of the Operator of the Anisotropy Perturbation of PbS and PbSe

Equations (71)

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H=|L6|L6|L6+|L6+Eg22kt22mt2kz22ml0mPlkzmPt(kxiky)0Eg22kt22mt2kz22mlmPt(kxiky)-mPlkzmPlkzmPt(kxiky)-Eg22kt22mt+2kz22ml+0mPt(kxiky)-mPlkz0-Eg22kt22mt+2kz22ml+,
Eg2+2kt22mt-+2kz22ml--E(k)×-Eg2-2kt22mt+-2kz22ml+-E(k)=2m2(Pt2kt2+Pl2kz2).
Hˆ0(k)=Eg2+2k22m-1Pmk·σPmk·σ-Eg2+2k22m+1.
3P2=2Pt2+Pl2,3/m±=2/mt±+1/ml±
Hˆ0(-i)F(r)=EF(r),
F(|r|=a)=0.
F(r)=[F1(r), F2(r), F3(r), F4(r)],
|Ψ(r)=F1(r)|L6-+F2(r)|L6-+F3(r)|L6++F4(r)|L6+.
Jˆ=Lˆ+2Σˆ,Σˆ=σ00σ,
ˆ=-Pˆ100Pˆ1.
Fπ,j,m(r)=ifl(r)l+m+1/22l+1 Ylm-1/2l-m+1/22l+1 Ylm+1/2fl+1(r)l-m+3/22l+3 Yl+1m-1/2-l+m+3/22l+3 Yl+1m+1/2,
j=l+1/2,π=(-1)l+1,
Fπ,j,m(r)=igl+1(r)l-m+3/22l+3 Yl+1m-1/2-l+m+3/22l+3 Yl+1m+1/2gl(r)l+m+1/22l+1 Ylm-1/2l-m+1/22l+1 Ylm+1/2,
j=(l+1)-1/2,π=(-1)l.
Eg2-E-22m-d2dr2+2rddr-l(l+1)r2fl(r)
-Pmddr+(l+2)rfl+1(r)=0,
Pmddr-lrfl(r)-Eg2+E-22m+d2dr2+2rddr
-(l+1)(l+2)r2fl+1(r)=0.
Eg2+E-22m+d2dr2+2rddr-l(l+1)r2gl(r)
-Pmddr+(l+2)rgl+1(r)=0,
Pmddr-lrgl(r)-Eg2-E-22m-d2dr2+2rddr
-(l+1)(l+2)r2gl+1(r)=0.
fl(r), gl(r)=ajl(kr)+bil(λr),
fl+1(r), gl+1(r)=cjl+1(kr)+dil+1(λr),
Eg2+2k22m--E(k)-Eg2-2k22m+-E(k)
=2m2P2k2,
Eg2-2λ22m--E(λ)-Eg2+2λ22m+-E(λ)
=-2m2P2λ2.
E±(k)=12[γk2±(Eg+αk2)2+β2k2],
α=221m-+1m+,β=2Pm,
γ=221m--1m+.
λ±(k)=2αEg+β2+(α2-γ2)k2+4γE±(k)α2-γ2.
ϱ±(k)jl+1(ka)il(λ±a)-μ±(k)jl(ka)il+1(λ±a)=0,
j=l+1/2,π=(-1)l+1,
ϱ±(k)jl(ka)il+1(λ±a)+μ±(k)jl+1(ka)il(λ±a)=0,
j=(l+1)-1/2,π=(-1)l.
ϱ±(k)=[Eg+(α+γ)k2-2E±(k)]/βk,
μ±(k)=[Eg-(α+γ)λ±(k)2-2E±(k)]/βλ±(k).
fl(r)=jl(kr)-jl(ka)il(λa)il(λr),
fl+1(r)=ϱ(k)jl+1(kr)-jl+1(ka)il+1(λa)il+1(λr),
gl+1(r)=jl+1(kr)-jl+1(ka)il+1(λa)il(λr),
gl(r)=-ϱ(k)jl(kr)-jl(ka)il(λa)il(λr).
π, j, m|Vˆ|π, j,m=π, j,-m|Vˆ|π, j,-m,
π, j, m|Vˆ|π, j-1, m=-π,j,-m|Vˆ|π,j-1,-m,
Mc,v=|Ψc(r)|e·p|Ψv(r)|2,
Mc,v=drFπc,jc,mc(r)(e·p)Fπv,jv,mv(r)+(e·zˆ)Pl×drFπc,jc,mc(r)(σxσz)Fπv,jv,mv(r)2.
Δj=0,±1,Δm=0,±1,andπcπv=-1.
HˆCoulomb=-c,c,v,vacbv˜bv˜acdr1dr2×Ψc(r1)Ψv(r2)×e2|r1-r2|Ψv(r2)Ψc(r1),
c,v|HˆCoulombLR|c,v=-dr1dr2Fc(r1)Fv(r2)×e2|r1-r2|Fv(r2)Fc(r1).
HˆExch=c,c,v,vacbv˜bv˜acdr1dr2Ψc(r1)Ψv(r2)×e2|r1-r2|Ψc(r2)Ψv(r1),
c,v|HˆExchLR|c,v=dr1dr2Fc(r1)Fv(r2)×e2|r1-r2|Fc(r2)Fv(r1).
J=1Ωunitcelldr1 1Ωunitcelldr2L6-(r1)|L6-(r2)×e2|r1-r2|L6+(r1)|L6+(r2),
|L6-|L6-|L6+|L6+
Vˆ=-T0(2)0δlT0(1)2δtT-1(1)0-T0(2)-2δtT1(1)-δlT0(1)δlT0(1)2δtT-1(1)+T0(2)0-2δtT1(1)-δlT0(1)0+T0(2),
T0(2)=2-32/z2.
T0(1)=/z,T±1(1)=/x±i/y2.
±=261ml±-1mt±,
δl,t=m(Pl,t-P).
π, j,m|Vˆ|π, j,m,
π, j,m|Vˆ|π, j-1,m,
π, j,m|Vˆ|π, j,m=- (2l-1)[l(l+2)+3/4-3m2]l(l+1)(2l+1)×fl(r)Yl0|T0(2)|fl(r)Yl0++ (2l+5)[l(l+2)+3/4-3m2](l+1)(l+2)(2l+3)×fl+1(r)Yl+10|T0(2)|fl+1(r)Yl+10+δl 4[l(l+2)+3/4-m2](l+1)(2l+1)(2l+3)×fl+1(r)Yl+10|T0(1)|fl(r)Yl0+δt 4[l(l+2)+3/4+m2](l+1)(2l+1)(2l+3)×fl+1(r)Yl+10|T0(1)|fl(r)Yl0.
π, j,m|Vˆ|π, j-1,m
=- 6ml(l+1)(2l+1)fl(r)Yl0|T0(2)|gl(r)Yl0++ 4m(l-m+1/2)(l+m+1/2)l(l+1)(2l-1)(2l+3)×fl+1(r)Yl+10|T0(2)|gl-1(r)Yl-10+(δt-δl) 2m(l-m+1/2)(l+m+1/2)l(2l-1)(2l+1)×fl(r)Yl0|T0(1)|gl-1(r)Yl-10+(δt-δl) 2m(l-m+1/2)(l+m+1/2)(l+1)(2l+1)(2l+3)×fl+1(r)Yl+10|T0(1)|gl(r)Yl0.
|Ψc,v(r)=F1c,v(r)|L6-+F2c,v(r)|L6-+F3c,v(r)|L6++F4c,v(r)|L6+i=14Fic,v(r)ui(r),
Mc,v=dri=14Ficui(e·p)j=14Fjvuj2=i,j=14Ωkcell[Fic(rk)]*(e·p)Fjv(rk)×1Ωunitcelldrui(r)uj(r)+Ωkcell[Fic(rk)]*Fjv(rk)×1Ωunitcelldrui(r)(e·p)uj(r)2,
1Ωunitcelldrui(r)uj(r)=δi,j,
1Ωunitcelldru1(r)pu3(r)=1Ωunitcelldru3(r)pu1(r)=Plzˆ,
1Ωunitcelldru2(r)pu4(r)=1Ωunitcelldru4(r)pu2(r)=-Plzˆ,
Mc,v=dri=14[Fic(r)]*(e·p)Fiv(r)+(e·zˆ)Pl{[F1c(r)]*F3v(r)+[F3c(r)]*F1v(r)-[F2c(r)]*F4v(r)-[F4c(r)]*F2v(r)}2,
c,v|HˆExch|c,v=dr1dr2Ψc(r1)Ψv(r2)×e2|r1-r2|Ψc(r2)Ψv(r1).
c,v|HˆExchLR|c,v=ijkldr1dr2[Fic(r1)]*Fjv(r1)×e2|r1-r2|[Fkv(r2)]*Flc(r2)×1Ωunitcelldrui(r)uj(r)×1Ωunitcelldruk(r)ul(r).

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