Abstract

We present a general approach to analyzing the optical activity of semiconductor nanocrystals of chiral shapes. By using a coordinate transformation that turns a chiral nanocrystal into a nanocuboid, we calculate the rotatory strengths, dissymmetry factors, and peak values of the circular dichroism (CD) signal upon intraband transitions inside the nanocrystal. It is shown that the atomic roughness of the nanocrystal surface can result in rotatory strengths as high as 10−36  erg×cm3 and in peak CD signals of about 0.1  cm−1 for typical nanocrystal densities of 1016  cm−3. The developed approach may prove useful for other nanocrystal shapes whereas the derived expressions apply directly for the modeling and interpretation of experimental CD spectra of quantum dots, nanorods, and nanoplatelets.

© 2016 Optical Society of America

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  1. J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
    [Crossref]
  2. F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
    [Crossref]
  3. M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
    [Crossref]
  4. M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
    [Crossref]
  5. A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
    [Crossref]
  6. A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
    [Crossref]
  7. J. D. Eshelby, J. Appl. Phys. 24, 176 (1953).
    [Crossref]
  8. S. D. Elliot, M. P. Moloney, and Y. K. Gun’ko, Nano Lett. 8, 2452 (2008).
    [Crossref]
  9. A. B. Migdal, Qualitative Methods in Quantum Theory (Da Capo, 2000).
  10. L. Rosenfeld, Z. Phys. 52, 161 (1929).
    [Crossref]
  11. A. S. Baimuratov, V. K. Turkov, I. D. Rukhlenko, and A. V. Fedorov, Opt. Lett. 37, 4645 (2012).
    [Crossref]
  12. N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
    [Crossref]
  13. N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
    [Crossref]

2016 (3)

F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
[Crossref]

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
[Crossref]

2015 (4)

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

2012 (1)

2008 (1)

S. D. Elliot, M. P. Moloney, and Y. K. Gun’ko, Nano Lett. 8, 2452 (2008).
[Crossref]

2005 (1)

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
[Crossref]

1953 (1)

J. D. Eshelby, J. Appl. Phys. 24, 176 (1953).
[Crossref]

1929 (1)

L. Rosenfeld, Z. Phys. 52, 161 (1929).
[Crossref]

Albelda, M. T.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
[Crossref]

Baimuratov, A. S.

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

A. S. Baimuratov, V. K. Turkov, I. D. Rukhlenko, and A. V. Fedorov, Opt. Lett. 37, 4645 (2012).
[Crossref]

Baranov, A. V.

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

Canary, J. W.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
[Crossref]

Elliot, S. D.

S. D. Elliot, M. P. Moloney, and Y. K. Gun’ko, Nano Lett. 8, 2452 (2008).
[Crossref]

Eshelby, J. D.

J. D. Eshelby, J. Appl. Phys. 24, 176 (1953).
[Crossref]

Fedorov, A. V.

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

A. S. Baimuratov, V. K. Turkov, I. D. Rukhlenko, and A. V. Fedorov, Opt. Lett. 37, 4645 (2012).
[Crossref]

Ginzburg, P.

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

Govan, J.

F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
[Crossref]

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

Gun’ko, Y. K.

F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
[Crossref]

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

S. D. Elliot, M. P. Moloney, and Y. K. Gun’ko, Nano Lett. 8, 2452 (2008).
[Crossref]

Leonov, M. Y.

N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
[Crossref]

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

Liu, Y.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
[Crossref]

Loudon, A.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

Maslov, V. G.

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

Migdal, A. B.

A. B. Migdal, Qualitative Methods in Quantum Theory (Da Capo, 2000).

Milton, F. P.

F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
[Crossref]

Moloney, M. P.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

S. D. Elliot, M. P. Moloney, and Y. K. Gun’ko, Nano Lett. 8, 2452 (2008).
[Crossref]

Mukhina, M.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

Mukhina, M. V.

F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

Noskov, R. E.

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

Orlova, A. O.

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

Ponomareva, I. O.

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

Purcell-Milton, F.

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

Rosenfeld, L.

L. Rosenfeld, Z. Phys. 52, 161 (1929).
[Crossref]

Rukhlenko, I. D.

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

A. S. Baimuratov, V. K. Turkov, I. D. Rukhlenko, and A. V. Fedorov, Opt. Lett. 37, 4645 (2012).
[Crossref]

Tepliakov, N. V.

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

N. V. Tepliakov, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, Opt. Express 24, A52 (2016).
[Crossref]

Turkov, V. K.

Zhang, J.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
[Crossref]

Chirality (1)

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, Chirality 17, 404 (2005).
[Crossref]

J. Appl. Phys. (1)

J. D. Eshelby, J. Appl. Phys. 24, 176 (1953).
[Crossref]

J. Phys. Chem. C (1)

N. V. Tepliakov, I. O. Ponomareva, M. Y. Leonov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, J. Phys. Chem. C 120, 2379 (2016).
[Crossref]

Nano Lett. (3)

S. D. Elliot, M. P. Moloney, and Y. K. Gun’ko, Nano Lett. 8, 2452 (2008).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Nano Lett. 15, 1710 (2015).
[Crossref]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, A. V. Fedorov, A. O. Orlova, F. Purcell-Milton, J. Govan, and Y. K. Gun’ko, Nano Lett. 15, 2844 (2015).
[Crossref]

Nanoscale Horiz. (1)

F. P. Milton, J. Govan, M. V. Mukhina, and Y. K. Gun’ko, Nanoscale Horiz. 1, 14 (2016).
[Crossref]

Nat. Protoc. (1)

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, Nat. Protoc. 10, 558 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Sci. Rep. (1)

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, Sci. Rep. 5, 14712 (2015).
[Crossref]

Z. Phys. (1)

L. Rosenfeld, Z. Phys. 52, 161 (1929).
[Crossref]

Other (1)

A. B. Migdal, Qualitative Methods in Quantum Theory (Da Capo, 2000).

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Figures (2)

Fig. 1.
Fig. 1. Coordinate transformation r = φ ( R ) turns a chiral semiconductor nanocrystal of irregular surface S ( R ) into a nanocuboid of the same volume and surface s ( r ) .

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Fig. 2.
Fig. 2. Three kinds of chiral nanocuboids whose optical activity can be described using the developed analytical approach: [(a) and (b)] two enantiomers with a pair of distorted facets ( β = γ = 0 ) ; [(c) and (d)] nanocuboids with two pairs of distorted facets ( γ = 0 ) ; and [(e)–(h)] nanocuboids with all facets distorted; L x = 6    nm , L y = 8    nm , L z = 10    nm , α = 0.06    nm − 1 , β = 0.08    nm − 1 , and γ = 0.1    nm − 1 .

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Tables (2)

Tables Icon

Table 1. Rotatory Strengths (in the Units of 10 − 38    erg × cm 3 ), Dissymmetry Factors, and Peak CD Signals (in cm − 1 ) for Three ED and Three MD Transitions Inside a Chiral ZnS Nanocrystal with L x = 3    nm , L y = 4    nm , L z = 5    nm , and α = 0.22    nm − 1

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Tables Icon

Table 2. Same as in Table 1 but for ED Transition | 111 ) → | 121 ) and MD Transition | 111 ) → | 212 ) inside Chiral ZnS Nanoplatelets, Nanorods, and Quantum Dots

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Equations (29)

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| n ) = | n ⟩ + σ nm | m ⟩ , E n = ε n + V nn ,
σ nm = ∑ m ≠ n V mn ε n − ε m .
⟨ r | n ⟩ = 8 / V 0 H x ( k x x ) H y ( k y y ) H z ( k z z ) ,
− L x + f ( Y , Z ) 2 ≤ X ≤ L x − f ( Y , Z ) 2 ,
− L y + g ( Z , X ) 2 ≤ Y ≤ L y − g ( Z , X ) 2 ,
− L z + h ( X , Y ) 2 ≤ Z ≤ L z − h ( X , Y ) 2 ,
| x | ≤ L x 2 , | y | ≤ L y 2 , | z | ≤ L z 2 ,
x = X + 1 2 f ( Y , Z ) ,
y = Y + 1 2 g ( Z , X ) ,
z = Z + 1 2 h ( X , Y ) .
δ V = F ( x , y , z ) ∂ y ∂ z + G ( x , y , z ) ∂ z ∂ x + H ( x , y , z ) ∂ x ∂ y ,
F ( x , y , z ) = ∂ g ∂ Z + ∂ h ∂ Y ,
G ( x , y , z ) = ∂ h ∂ X + ∂ f ∂ Z ,
H ( x , y , z ) = ∂ f ∂ Y + ∂ g ∂ X .
R mn ED = − i ρ r nm ( σ ms L sn − σ ns L sm ) ,
R mn MD = − i ρ L mn ( σ ns r sm + σ ms r sn ) ,
r nm = ( L x B n x m x δ n y m y δ n z m z L y δ n x m x B n y m y δ n z m z L z δ n x m x δ n y m y B n z m z ) , L sn i ℏ = ( δ n x s x G n y s y n z s z δ n y s y G n z s z n x s x δ n z s z G n x s x n y s y ) ,
B n x m x = − 1 π 2 8 n x m x ( n x 2 − m x 2 ) 2   sin [ π 2 ( n x + m x ) ] ,
G n y s y n z s z = 1 2 L y L z B n y s y B n z s z ( k n y 2 − k s y 2 − k n z 2 + k s z 2 ) ,
F ( x , y , z ) = − F ( − x , y , z ) = F ( x , − y , z ) = F ( x , y , − z ) ,
G ( x , y , z ) = G ( − x , y , z ) = − G ( x , − y , z ) = G ( x , y , − z ) ,
H ( x , y , z ) = H ( − x , y , z ) = H ( x , − y , z ) = − H ( x , y , − z ) .
X ≈ x − ( α / 2 ) y z ,
Y ≈ y − ( β / 2 ) z x ,
Z ≈ z − ( γ / 2 ) x y .
δ V = α ∂ x ( y ∂ z + z ∂ y ) + β ∂ y ( z ∂ x + x ∂ z ) + γ ∂ z ( x ∂ y + y ∂ x ) .
V ms = 1 4 V 0 B m x s x B m y s y B m z s z × [ α ( k s x 2 − k m x 2 ) ( k s y 2 − k m y 2 + k s z 2 − k m z 2 ) + β ( k s y 2 − k m y 2 ) ( k s z 2 − k m z 2 + k s x 2 − k m x 2 ) + γ ( k s z 2 − k m z 2 ) ( k s x 2 − k m x 2 + k s y 2 − k m y 2 ) ] .
g mn ED = 4 R mn ED / ( e r nm ) 2 ,
g mn MD = 4 R mn MD ( ρ / e ) 2 L mn 2 + e 2 ( σ ns r sm + σ ms r sn ) 2 .

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