Abstract

Metallic spherical dome shells have received much attention in recent years because they have proven to possess highly impressive optical properties. The expected distinctive changes occurring owing to quantum confinement of conduction electrons in these nanoparticles as their thickness is reduced, have not been properly investigated. Here we carry out a detailed analytical derivation of the quantum contributions by introducing linearly shifted Associated Legendre Polynomials, which form an approximate orthonormal eigenbasis for the single-electron Hamiltonian of a spherical dome shell. Our analytical results clearly show the contribution of different elements of a spherical dome shell to the effective dielectric function. More specifically, our results provide an accurate, quantitative correction for the dielectric function of metallic spherical dome shells with thickness below 10 nm.

© 2014 Optical Society of America

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2013 (6)

D. Sikdar, I. D. Rukhlenko, W. Cheng, and M. Premaratne, “Optimized gold nanoshell ensembles for biomedical applications,” Nanoscale Res. Lett. 8, 1–5 (2013).
[Crossref]

X. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

C. Rupasinghe, I. D. Rukhlenko, and M. Premaratne, “Design optimization of spasers considering the degeneracy of excited plasmon modes,” Opt. Express 21, 15335–15349 (2013).
[Crossref] [PubMed]

N. S. King, M. W. Knight, N. Large, A. M. Goodman, P. Nordlander, and N. J. Halas, “Orienting nanoantennas in three dimensions to control light scattering across a dielectric interface,” Nano Lett. 13, 5997–6001 (2013).
[Crossref] [PubMed]

R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Competition between surface screening and size quantization for surface plasmons in nanoparticles,” New J. Phys. 15, 083044 (2013).
[Crossref]

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with eels,” Nanophotonics 2, 131–138 (2013).
[Crossref]

2012 (1)

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref] [PubMed]

2011 (3)

P. Van Dorpe and J. Ye, “Semishells: versatile plasmonic nanoparticles,” ACS Nano 5, 6774–6778 (2011).
[Crossref] [PubMed]

Y. Zhang, A. Barhoumi, J. B. Lassiter, and N. J. Halas, “Orientation-preserving transfer and directional light scattering from individual light-bending nanoparticles,” Nano Lett. 11, 1838–1844 (2011).
[Crossref] [PubMed]

N. S. King, Y. Li, C. Ayala-Orozco, T. Brannan, P. Nordlander, and N. J. Halas, “Angle-and spectral-dependent light scattering from plasmonic nanocups,” ACS Nano 5, 7254–7262 (2011).
[Crossref] [PubMed]

2010 (4)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

V. E. Ferry, M. A. Verschuuren, H. B. Li, E. Verhagen, R. J. Walters, R. E. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18, A237–A245 (2010).
[Crossref] [PubMed]

Y. He and T. Zeng, “First-principles study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C 114, 18023–18030 (2010).
[Crossref]

J. Ye, N. Verellen, W. Van Roy, L. Lagae, G. Maes, G. Borghs, and P. Van Dorpe, “Plasmonic modes of metallic semishells in a polymer film,” ACS Nano 4, 1457–1464 (2010).
[Crossref] [PubMed]

2009 (3)

J. Ye, L. Lagae, G. Maes, G. Borghs, and P. V. Dorpe, “Symmetry breaking induced optical properties of gold open shell nanostructures,” Opt. Express 17, 23765–23771 (2009).
[Crossref]

N. A. Mirin and N. J. Halas, “Light-bending nanoparticles,” Nano Lett. 9, 1255–1259 (2009).
[Crossref] [PubMed]

J. Ye, P. Van Dorpe, W. Van Roy, K. Lodewijks, I. De Vlaminck, G. Maes, and G. Borghs, “Fabrication and optical properties of gold semishells,” J. Phys. Chem. C 113, 3110–3115 (2009).
[Crossref]

2007 (1)

M. Cortie and M. Ford, “A plasmon-induced current loop in gold semi-shells,” Nanotechnology 18, 235704 (2007).
[Crossref]

2006 (1)

R. Chang and P. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
[Crossref]

2005 (1)

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5, 119–124 (2005).
[Crossref] [PubMed]

2004 (2)

X. Xu, M. Stevens, and M. Cortie, “In situ precipitation of gold nanoparticles onto glass for potential architectural applications,” Chem. Mater. 16, 2259–2266 (2004).
[Crossref]

E. Prodan and P. Nordlander, “Plasmon hybridization in spherical nanoparticles,” J. Chem. Phys. 120, 5444–5454 (2004).
[Crossref] [PubMed]

2003 (5)

A. T. Tilke, F. C. Simmel, H. Lorenz, R. H. Blick, and J. P. Kotthaus, “Quantum interference in a one-dimensional silicon nanowire,” Phys. Rev. B 68, 075311 (2003).
[Crossref]

S. Bruzzone, G. Arrighini, and C. Guidotti, “Some spectroscopic properties of gold nanorods according to a schematic quantum model founded on the dielectric behavior of the electron-gas confined in a box. I,” Chem. Phys. 291, 125–140 (2003).
[Crossref]

C. Charnay, A. Lee, S. Q. Man, C. E. Moran, C. Radloff, R. K. Bradley, and N. J. Halas, “Reduced symmetry metallodielectric nanoparticles: chemical synthesis and plasmonic properties,” J. Phys. Chem. B 107, 7327–7333 (2003).
[Crossref]

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe (core/shell) and CdSe/ZnTe (core/shell) heterostructures,” J. Am. Chem. Soc. 125, 11466–11467 (2003).
[Crossref] [PubMed]

E. Prodan, P. Nordlander, and N. Halas, “Electronic structure and optical properties of gold nanoshells,” Nano Lett. 3, 1411–1415 (2003).
[Crossref]

2002 (1)

S. Westcott, J. Jackson, C. Radloff, and N. Halas, “Relative contributions to the plasmon line shape of metal nanoshells,” Phys. Rev. B 66, 155431 (2002).
[Crossref]

1999 (2)

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824–1832 (1999).
[Crossref]

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. B 103, 8410–8426 (1999).
[Crossref]

1996 (1)

N. Garcia and J. Costa-Krämer, “Quantum-level phenomena in nanowires,” Europhys. News 27, 89–91 (1996).

1992 (1)

F. Ouyang, P. Batson, and M. Isaacson, “Quantum size effects in the surface-plasmon excitation of small metallic particles by electron-energy-loss spectroscopy,” Phys. Rev. B 46, 15421 (1992).
[Crossref]

1983 (1)

W. Kraus and G. C. Schatz, “Plasmon resonance broadening in small metal particles,” J. Chem. Phys. 79, 6130–6139 (1983).
[Crossref]

1975 (1)

L. Genzel, T. Martin, and U. Kreibig, “Dielectric function and plasma resonances of small metal particles,” Z. Phys. B 21, 339–346 (1975).
[Crossref]

1972 (1)

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1966 (1)

A. Kawabata and R. Kubo, “Electronic properties of fine metallic particles. II. plasma resonance absorption,” J. Phys. Soc. Jpn. 21, 1765–1772 (1966).
[Crossref]

Agrawal, G.

M. Premaratne and G. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University, 2011).
[Crossref]

Antosiewicz, T. J.

R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Competition between surface screening and size quantization for surface plasmons in nanoparticles,” New J. Phys. 15, 083044 (2013).
[Crossref]

Apell, S. P.

R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Competition between surface screening and size quantization for surface plasmons in nanoparticles,” New J. Phys. 15, 083044 (2013).
[Crossref]

Arrighini, G.

S. Bruzzone, G. Arrighini, and C. Guidotti, “Some spectroscopic properties of gold nanorods according to a schematic quantum model founded on the dielectric behavior of the electron-gas confined in a box. I,” Chem. Phys. 291, 125–140 (2003).
[Crossref]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

V. E. Ferry, M. A. Verschuuren, H. B. Li, E. Verhagen, R. J. Walters, R. E. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18, A237–A245 (2010).
[Crossref] [PubMed]

Averitt, R. D.

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824–1832 (1999).
[Crossref]

Ayala-Orozco, C.

N. S. King, Y. Li, C. Ayala-Orozco, T. Brannan, P. Nordlander, and N. J. Halas, “Angle-and spectral-dependent light scattering from plasmonic nanocups,” ACS Nano 5, 7254–7262 (2011).
[Crossref] [PubMed]

Barhoumi, A.

Y. Zhang, A. Barhoumi, J. B. Lassiter, and N. J. Halas, “Orientation-preserving transfer and directional light scattering from individual light-bending nanoparticles,” Nano Lett. 11, 1838–1844 (2011).
[Crossref] [PubMed]

Batson, P.

F. Ouyang, P. Batson, and M. Isaacson, “Quantum size effects in the surface-plasmon excitation of small metallic particles by electron-energy-loss spectroscopy,” Phys. Rev. B 46, 15421 (1992).
[Crossref]

Bawendi, M.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe (core/shell) and CdSe/ZnTe (core/shell) heterostructures,” J. Am. Chem. Soc. 125, 11466–11467 (2003).
[Crossref] [PubMed]

Blick, R. H.

A. T. Tilke, F. C. Simmel, H. Lorenz, R. H. Blick, and J. P. Kotthaus, “Quantum interference in a one-dimensional silicon nanowire,” Phys. Rev. B 68, 075311 (2003).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

Borghs, G.

J. Ye, N. Verellen, W. Van Roy, L. Lagae, G. Maes, G. Borghs, and P. Van Dorpe, “Plasmonic modes of metallic semishells in a polymer film,” ACS Nano 4, 1457–1464 (2010).
[Crossref] [PubMed]

J. Ye, P. Van Dorpe, W. Van Roy, K. Lodewijks, I. De Vlaminck, G. Maes, and G. Borghs, “Fabrication and optical properties of gold semishells,” J. Phys. Chem. C 113, 3110–3115 (2009).
[Crossref]

J. Ye, L. Lagae, G. Maes, G. Borghs, and P. V. Dorpe, “Symmetry breaking induced optical properties of gold open shell nanostructures,” Opt. Express 17, 23765–23771 (2009).
[Crossref]

Bradley, R. K.

C. Charnay, A. Lee, S. Q. Man, C. E. Moran, C. Radloff, R. K. Bradley, and N. J. Halas, “Reduced symmetry metallodielectric nanoparticles: chemical synthesis and plasmonic properties,” J. Phys. Chem. B 107, 7327–7333 (2003).
[Crossref]

Brannan, T.

N. S. King, Y. Li, C. Ayala-Orozco, T. Brannan, P. Nordlander, and N. J. Halas, “Angle-and spectral-dependent light scattering from plasmonic nanocups,” ACS Nano 5, 7254–7262 (2011).
[Crossref] [PubMed]

Bruzzone, S.

S. Bruzzone, G. Arrighini, and C. Guidotti, “Some spectroscopic properties of gold nanorods according to a schematic quantum model founded on the dielectric behavior of the electron-gas confined in a box. I,” Chem. Phys. 291, 125–140 (2003).
[Crossref]

Burrows, A.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with eels,” Nanophotonics 2, 131–138 (2013).
[Crossref]

Chang, R.

R. Chang and P. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
[Crossref]

Charnay, C.

C. Charnay, A. Lee, S. Q. Man, C. E. Moran, C. Radloff, R. K. Bradley, and N. J. Halas, “Reduced symmetry metallodielectric nanoparticles: chemical synthesis and plasmonic properties,” J. Phys. Chem. B 107, 7327–7333 (2003).
[Crossref]

Cheng, W.

D. Sikdar, I. D. Rukhlenko, W. Cheng, and M. Premaratne, “Optimized gold nanoshell ensembles for biomedical applications,” Nanoscale Res. Lett. 8, 1–5 (2013).
[Crossref]

Christy, R.-W.

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cortie, M.

M. Cortie and M. Ford, “A plasmon-induced current loop in gold semi-shells,” Nanotechnology 18, 235704 (2007).
[Crossref]

X. Xu, M. Stevens, and M. Cortie, “In situ precipitation of gold nanoparticles onto glass for potential architectural applications,” Chem. Mater. 16, 2259–2266 (2004).
[Crossref]

Costa-Krämer, J.

N. Garcia and J. Costa-Krämer, “Quantum-level phenomena in nanowires,” Europhys. News 27, 89–91 (1996).

De Vlaminck, I.

J. Ye, P. Van Dorpe, W. Van Roy, K. Lodewijks, I. De Vlaminck, G. Maes, and G. Borghs, “Fabrication and optical properties of gold semishells,” J. Phys. Chem. C 113, 3110–3115 (2009).
[Crossref]

Dionne, J. A.

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref] [PubMed]

Dorpe, P. V.

J. Ye, L. Lagae, G. Maes, G. Borghs, and P. V. Dorpe, “Symmetry breaking induced optical properties of gold open shell nanostructures,” Opt. Express 17, 23765–23771 (2009).
[Crossref]

Eisler, H.-J.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe (core/shell) and CdSe/ZnTe (core/shell) heterostructures,” J. Am. Chem. Soc. 125, 11466–11467 (2003).
[Crossref] [PubMed]

El-Sayed, M. A.

S. Link and M. A. El-Sayed, “Spectral properties and relaxation dynamics of surface plasmon electronic oscillations in gold and silver nanodots and nanorods,” J. Phys. Chem. B 103, 8410–8426 (1999).
[Crossref]

Ferry, V. E.

V. E. Ferry, M. A. Verschuuren, H. B. Li, E. Verhagen, R. J. Walters, R. E. Schropp, H. A. Atwater, and A. Polman, “Light trapping in ultrathin plasmonic solar cells,” Opt. Express 18, A237–A245 (2010).
[Crossref] [PubMed]

Fischer, S. V.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with eels,” Nanophotonics 2, 131–138 (2013).
[Crossref]

Fisher, B.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe (core/shell) and CdSe/ZnTe (core/shell) heterostructures,” J. Am. Chem. Soc. 125, 11466–11467 (2003).
[Crossref] [PubMed]

Ford, M.

M. Cortie and M. Ford, “A plasmon-induced current loop in gold semi-shells,” Nanotechnology 18, 235704 (2007).
[Crossref]

Fujita, K.

X. Meng, U. Guler, A. V. Kildishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
[Crossref] [PubMed]

Garcia, N.

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J. Ye, N. Verellen, W. Van Roy, L. Lagae, G. Maes, G. Borghs, and P. Van Dorpe, “Plasmonic modes of metallic semishells in a polymer film,” ACS Nano 4, 1457–1464 (2010).
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J. Ye, P. Van Dorpe, W. Van Roy, K. Lodewijks, I. De Vlaminck, G. Maes, and G. Borghs, “Fabrication and optical properties of gold semishells,” J. Phys. Chem. C 113, 3110–3115 (2009).
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X. Xu, M. Stevens, and M. Cortie, “In situ precipitation of gold nanoparticles onto glass for potential architectural applications,” Chem. Mater. 16, 2259–2266 (2004).
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P. Van Dorpe and J. Ye, “Semishells: versatile plasmonic nanoparticles,” ACS Nano 5, 6774–6778 (2011).
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Y. He and T. Zeng, “First-principles study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C 114, 18023–18030 (2010).
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Y. Zhang, A. Barhoumi, J. B. Lassiter, and N. J. Halas, “Orientation-preserving transfer and directional light scattering from individual light-bending nanoparticles,” Nano Lett. 11, 1838–1844 (2011).
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ACS Nano (3)

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Figures (4)

Fig. 1
Fig. 1 Cross section of a spherical dome shell along the xz plane with the center on the origin. The axis of symmetry is along the z direction of a right-handed Cartesian coordinate system.
Fig. 2
Fig. 2 Imaginary (a) and real (b) parts of the dielectric functions of spherical dome shells of varying thickness calculated with the quantum mechanical model are shown by solid lines. Cut-off angle is 25.8° (ζ = 0.9) for the spherical dome shells. Empirical values obtained from Drude model with size dependent correction for a shell with a = 40nm and b = 45nm is shown by the dotted-dashed line. a and b values given in the figure are in nanometers. The dashed line and dotted line shows the experimental bulk dielectric values taken from Refs. [27] and [38] respectively.
Fig. 3
Fig. 3 Imaginary (a) and real (b) parts of the dielectric function of spherical dome shells for four different cut-off angles (solid lines). The dielectric function of a complete shell (θ′ = 0) calculated with the quantum-mechanical model is shown by a dashed line, while values obtained from the Drude model with size dependent corrections ares shown by a dotted dash line. In all calculations a = 40nm and b = 43nm.
Fig. 4
Fig. 4 Extinction cross section and the imaginary part of the dielectric functions of a spherical dome shell calculated using our model and the Drude model with size dependent correction. The dimensions are a = 40nm, b = 45nm and cut-off angle = 25.8° (ζ = 0.9). For extinction cross section calculations, the core of the spherical dome shell was taken as Si (ε = 2.04) and the surrounding as air (ε = 1).

Equations (71)

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ε ( ω ) = ε inter ( ω ) + ε intra ( ω ) ,
ε intra ( ω ) = 1 ω p 2 ω 2 + j ω γ ,
ω p = n e q e 2 ε 0 μ ,
γ = γ bulk + A v f L eff ,
ε = [ ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz ] .
ε zz ( ω ) = 1 + P zz ( ω ) ε 0 z sin ω t .
H 0 ψ i ( r ) = E i ψ i ( r ) ,
j h ¯ ψ f ( r , t ) t = [ H 0 + H ( t ) ] ψ f ( r , t ) ,
H ( t ) = q e z z sin ω t ,
ψ f ( r , t ) = 1 2 q e z i ψ i ( r ) | z | ψ f ( r ) ψ f ( r ) exp ( j E f t h ¯ ) × { 1 exp [ j ( ω i , f + ω ) t ] ω i , f + ω + 1 exp [ j ( ω i , f ω ) t ] ω i , f ω } ,
ω i , f = ( E f E i ) / h ¯ .
p ¯ f zz ( ω ) = ψ f ( r , t ) | q e z | ψ f ( r , t ) .
P zz ( ω ) = 1 V p f p ¯ f z z ( ω ) ,
ε z z ( ω ) = 1 + 2 q e 2 ε 0 V p h ¯ i f | ψ i ( r ) | z | ψ f ( r ) | 2 ω i f ω i , f 2 ω 2 j γ ω .
ε zz ( ω ) = ε inter zz ( ω ) + 1 + ω p 2 i f S i , f zz ω i , f 2 ω 2 j γ ω ,
S i , f zz = 2 μ ω i , f h ¯ N | ψ f | z | ψ i | 2 ,
[ ε zz ( ω ) ] im = [ ε inter zz ( ω ) ] im + ω p 2 i f S i , f zz γ ω ( ω i , f 2 ω 2 ) 2 + γ 2 ω 2 .
[ ε zz ( ω ) ] re = 2 ω π P 0 [ ε zz ( ω ) ] im 1 ω 2 ω 2 d ω ,
[ ε zz ( ω ) ] im = [ ε inter zz ( ω ) ] im + ω p 2 Δ ν S Δ ν zz γ ω ( Ω Δ ν zz 2 ω 2 ) 2 + γ 2 ω 2 ,
S Δ ν z z = i f Δ ν S i , f zz ,
Ω Δ ν zz = 1 S Δ ν z z i f Δ ν S i , f zz ω i , f .
h ¯ 2 2 μ 1 r 2 sin θ [ sin θ r ( r 2 ψ r ) + θ ( sin θ ψ θ ) + 1 sin θ 2 ψ ϕ 2 ] + V ( r ) ψ = E ψ ,
V ( r ) = { 0 if a r b otherwise .
ψ κ ( r , θ , ϕ ) [ ψ p ( r , θ , ϕ ) ] * d 3 V p = δ κ , ρ ,
ψ ( r , θ , ϕ ) = R ( r ) Q ( θ ) F ( ϕ ) .
a b R κ ( r ) R ρ * ( r ) r 2 d r = δ κ , ρ
0 2 π θ π F κ ( ϕ ) F ρ * Q κ ( θ ) Q ρ * ( θ ) sin θ d θ d ϕ = δ κ , ρ .
d 2 R ( r ) d r 2 + 2 r d R ( r ) d r + [ 2 m E h ¯ 2 l ( l + 1 ) r 2 ] R ( r ) = 0 ,
R ( r ) = 1 r 2 b a sin [ n π b a ( r a ) ]
E n , l = h 2 2 μ [ n 2 π 2 ( b a ) 2 + l 2 a 2 ] ,
d 2 F ( ϕ ) d ϕ 2 + m 2 F ( ϕ ) = 0 ,
F ( ϕ ) = F ( ϕ + 2 π ) ,
F ( ϕ ) = 1 2 π exp ( j m ϕ ) ,
sin θ d d θ [ sin θ d Q d θ ] + [ l ( l + 1 ) sin 2 θ m 2 ] Q = 0 ,
( 1 x 2 ) d 2 Q d x 2 2 x d Q d x + [ l ( l + 1 ) m 2 1 x 2 ] Q = 0 ,
1 x ζ , Q ( ζ ) = 0 .
P l m ( x ) = ( 1 ) l 2 l l ! ( 1 x 2 ) m 2 d l + m d x l + m ( 1 x 2 ) l .
1 1 P l m ( x ) P l m ( x ) d x = 2 ( m + l ) ! ( 2 l + 1 ) ( l m ) ! δ l , l ,
( l m ) P l m ( x ) = x ( 2 l 1 ) P l 1 m ( x ) ( l + m 1 ) P l 2 m ( x ) ,
P l m ( x ) = ( 1 ) m ( l m ) ! ( l + m ) ! P l m ( x ) .
Q ( x ) = ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! P l m ( x ) , l > 0 , l m l .
P ˜ l m ( x ) = P l m ( k 1 x + k 2 ) ,
k 1 = 2 ζ + 1 , k 2 = 1 ζ ζ + 1 .
Q ( x ) = k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! P ˜ l m ( x ) , l > 0 , l m l .
1 ζ Q ˜ l m ( x ) Q ˜ l m ( x ) d x = 1 .
ψ n , l , m ( r , θ , ϕ ) = k 1 ( 2 l + 1 ) ( l m ) ! 2 π ( b a ) ( m + l ) ! 1 r sin [ n π b a ( r a ) ] P ˜ l m ( cos θ ) exp ( j m ϕ ) ,
M i f = 0 2 π d ϕ θ π sin θ d θ a b r 2 d r ψ ( r , θ , ϕ ) n , l , m r cos θ [ ψ ( r , θ , ϕ ) n + Δ n , l + Δ l , m + Δ m ] *
M i f = δ Δ m , 0 [ ( l + 1 + m ) ( l + 1 m ) ( 2 l + 1 ) ( 2 l + 3 ) δ Δ l , 1 k 1 + ( l + m ) ( l m ) ( 2 l 1 ) ( 2 l + 1 ) δ Δ l , 1 k 1 k 2 k 1 δ Δ l , 0 ] × { 4 ( b a ) [ ( 1 ) Δ n 1 ] n ( n + Δ n ) π 2 ( 2 n + Δ n ) 2 Δ n 2 + b δ Δ n , 0 } .
ω i , f = h ¯ 2 μ [ Δ n ( 2 n + Δ n ) π 2 ( b a ) 2 + Δ l ( 2 l + Δ l + 1 ) a 2 ] .
E Fermi = h ¯ 2 2 μ [ n f 2 π 2 ( b a ) 2 + 1 a 2 ] h ¯ 2 2 μ [ n f 2 π 2 ( b a ) 2 ] ,
l l max = Int [ ( n f 2 n 2 ) π a ( b a ) ] ,
N = n = 1 n f l = 1 l max l m l m 0 2 = 2 n = 1 n f l max ( l max + 1 ) 4 π 2 a 2 n f 3 3 ( b a ) 2 .
S i , f zz = S n , Δ n , l , Δ l , m , Δ m zz 3 ( b a ) 2 4 π 2 a 2 n f 3 [ Δ n ( 2 n + Δ n ) π 2 ( b a ) 2 + Δ l ( 2 l + Δ l + 1 ) a 2 ] × δ Δ m , 0 [ ( l + 1 + m ) ( l + 1 m ) ( 2 l + 1 ) ( 2 l + 3 ) δ Δ l , 1 k 1 2 + ( l + m ) ( l m ) ( 2 l 1 ) ( 2 l + 1 ) δ Δ l , 1 k 1 2 + k 2 2 k 1 2 δ Δ l , 0 ] × { 16 ( b a ) 2 [ ( 1 ) Δ n 1 ] 2 n 2 ( n + Δ n ) 2 π 4 ( 2 n + Δ n ) 4 Δ n 4 + b 2 δ Δ n , 0 } .
l > g = ceil [ π a ( b a ) n f 2 ( n + Δ n ) 2 Δ l ] ,
S Δ n zz = n = 1 n max Δ l = 1 1 l = l min l max l m l m 0 Δ m S n , Δ n , l , Δ l , m , Δ m zz .
S Δ n zz = n = 1 n max Δ l = 1 1 l = l min l max 3 ( b a ) 2 4 π 2 k 1 2 a 2 n f 3 { l ( 5 + 9 l + 4 l 2 ) 3 ( 3 + 8 l + 4 l 2 ) [ H + 2 ( l + 1 ) ( b a ) 2 H π 2 a 2 ( 2 n + Δ n ) Δ n + 2 ( l + 1 ) b 2 a 2 δ Δ n , 0 ] δ Δ l , 1 + l ( 1 + 3 l 4 l 2 ) 3 ( 1 4 l 2 ) [ H 2 l ( b a ) 2 H π 2 a 2 ( 2 n + Δ n ) Δ n ] δ Δ l , 1 + 2 l k 2 2 H δ Δ l , 0 } ,
H ( n , Δ n ) = 16 [ ( 1 ) Δ n 1 ] 2 n 2 ( n + Δ n ) 2 π 2 ( 2 n + Δ n ) 3 Δ n 3 .
Ω Δ n zz = 1 S Δ n zz n = 1 n max Δ l = 1 1 l = l min l max S Δ n zz h ¯ 2 μ [ Δ n ( 2 n + Δ n ) π 2 ( b a ) 2 + Δ l ( 2 l + Δ l + 1 ) a 2 ] .
ε inter ( ω ) = 3.66 1 exp ( 4.08 h ¯ ω / q e ) 1 .
1 ζ Q ˜ l m ( x ) Q ˜ l m ( x ) d x = 1 ζ k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! P ˜ l m ( x ) k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! P ˜ l m ( x ) d x = k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! 1 k 1 1 1 P l m ( x ´ ) P l m ( x ´ ) d x ´ = k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! 1 k 1 2 ( m + l ) ! ( 2 l + 1 ) ( l m ) ! δ l , l = k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! 1 k 1 2 ( m + l ) ! ( 2 l + 1 ) ( l m ) ! δ l , l = 1 .
( l m ) P l m ( k 1 x + k 2 ) = ( k 1 x + k 2 ) ( 2 l 1 ) P l 1 m ( k 1 x + k 2 ) ( l + m 1 ) P l 2 m ( k 1 x + k 2 ) ( l m ) P ˜ l m ( x ) = ( k 1 x + k 2 ) ( 2 l 1 ) P ˜ l 1 m ( x ) ( l + m 1 ) P ˜ l 2 m ( x ) ( k 1 x + k 2 ) ( 2 l 1 ) P ˜ l 1 m ( x ) = ( l m ) P ˜ l m ( x ) + ( l + m 1 ) P ˜ l 2 m ( x ) k 2 ( 2 l 1 ) P ˜ l 1 m ( x ) k 1 ( 2 l 1 ) .
M i f = 0 2 π θ π { sin θ cos θ Q ( θ ) l , m [ Q ( θ ) l + Δ l , m + Δ m ] * F ( ϕ ) m [ F ( ϕ ) m + Δ m ] * d θ d ϕ } × a b r 3 R ( r ) n , l [ R ( r ) n + Δ n , l + Δ l ] * d r .
a b r 3 R ( r ) n , l [ R ( r ) n + Δ n , l + Δ l ] * d r = 4 ( b a ) [ ( 1 ) Δ n 1 ] n ( n + Δ n ) π 2 ( 2 n + Δ n ) 2 Δ n 2 + b δ Δ n , 0 .
0 2 π θ π sin θ cos θ Q ( θ ) l , m [ Q ( θ ) l + Δ l , m + Δ m ] * F ( ϕ ) m [ F ( ϕ ) m + Δ m ] * d θ d ϕ = k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! k 1 [ 2 ( l + Δ l ) + 1 ] [ ( l + Δ l ) ( m + Δ m ) ] ! 2 [ ( m + Δ m ) + ( l + Δ l ) ] ! 0 2 π 1 2 π exp ( j m ϕ ) 1 2 π exp [ i ( m + Δ m ) ϕ ] d ϕ θ π sin θ cos θ P ˜ l m ( cos θ ) P ˜ l + Δ l m + Δ m ( cos θ ) d θ = δ Δ m , 0 k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! k 1 [ 2 ( l + Δ l ) + 1 ] [ ( l + Δ l ) m ] ! 2 [ m + ( l + Δ l ) ] ! θ π sin θ cos θ P ˜ l m ( cos θ ) P ˜ l + Δ l m ( cos θ ) d θ = δ Δ m , 0 k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! k 1 [ 2 ( l + Δ l ) ] [ ( l + Δ l ) m ] ! 2 [ m + ( l + Δ l ) ] ! 1 ζ x P ˜ l m ( x ) P ˜ l + Δ l m ( x ) d x .
0 2 π θ π sin θ cos θ Q ( θ ) l , m [ Q ( θ ) l + Δ l , m + Δ m ] * F ( ϕ ) m [ F ( ϕ ) m + Δ m ] * d θ d ϕ = δ Δ m , 0 k 1 ( 2 l + 1 ) ( l m ) ! 2 ( m + l ) ! k 1 [ 2 ( l + Δ l ) ] [ ( l + Δ l ) m ] ! 2 [ m + ( l + Δ l ) ] ! × 1 ζ { [ ( l + 1 ) m ] P ˜ l + 1 m ( x ) + [ ( l + 1 ) + m 1 ] P ˜ ( l + 1 ) 2 m ( x ) k 2 [ 2 ( l + 1 ) 1 ] P ˜ ( l + 1 ) 1 m ( x ) k 1 [ 2 ( l + 1 ) 1 ] × P ˜ l + Δ l m ( x ) } d x = δ Δ m , 0 [ ( l + 1 + m ) ( l + 1 m ) ( 2 l + 1 ) ( 2 l + 3 ) δ Δ l , 1 k 1 + ( l + m ) ( l m ) ( 2 l 1 ) ( 2 l + 1 ) δ Δ l , 1 k 1 k 2 k 1 δ Δ l , 0 ] .
M i f = { 4 ( b a ) [ ( 1 ) Δ n 1 ] n ( n + Δ n ) π 2 ( 2 n + Δ n ) 2 Δ n 2 + b δ Δ n , 0 } × δ Δ m , 0 [ ( l + 1 + m ) ( l + 1 m ) ( 2 l + 1 ) ( 2 l + 3 ) δ Δ l , 1 k 1 + ( l + m ) ( l m ) ( 2 l 1 ) ( 2 l + 1 ) δ Δ l , 1 k 1 k 2 k 1 δ Δ l , 0 ]
S Δ n zz = n = 1 n max l = l min l max Δ l = 1 1 ( { 4 ( b a ) 2 [ ( 1 ) Δ n 1 ] 2 n 2 ( n + Δ n ) 2 ( l + 1 ) π 4 a 2 n f 3 ( 2 n + Δ n ) 3 Δ n 3 + 8 ( b a ) 4 [ ( 1 ) Δ n 1 ] 2 n 2 ( n + Δ n ) 2 ( l + 1 ) 2 π 6 a 4 n f 3 ( 2 n + Δ n ) 4 Δ n 4 + ( b a ) 2 ( l + 1 ) 2 a 2 π 2 n f 3 δ Δ n , 0 } δ Δ l , + 1 + { 4 ( b a ) 2 [ ( 1 ) Δ n 1 ] 2 n 2 ( n + Δ n ) 2 l π 4 a 2 n f 3 ( 2 n + Δ n ) 3 Δ n 3 8 ( b a ) 4 [ ( 1 ) Δ n 1 ] 2 n 2 ( n + Δ n ) 2 l 2 π 6 a 4 n f 3 ( 2 n + Δ n ) 4 Δ n 4 } δ Δ l , 1 )
Ω Δ n zz = 1 S Δ n zz n = 1 n max l = l min l max Δ l = 1 1 S Δ n zz h ¯ 2 μ [ Δ n ( 2 n + Δ n ) π 2 ( b a ) 2 + Δ l ( 2 l + Δ l + 1 ) a 2 ] .
n max = [ [ 2 μ h ¯ 2 [ ( b a ) 2 E Fermi π 2 ] ] ] ,
l max = [ π a ( b a ) ( n f 2 n 2 ) ] ,
l min = max { 0 , [ a π ( b a ) n f 2 ( n + Δ n ) 2 Δ l ] } .

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