Optically Active Semiconductor Nanosprings for Tunable Chiral

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Optically Active Semiconductor Nanosprings for Tunable Chiral Nanophotonics Anvar S. Baimuratov, Tatiana P. Pereziabova, Mikhail Yu. Leonov, Weiren Zhu, Alexander V. Baranov, Anatoly V. Fedorov, Yurii K. Gun'ko, and Ivan D. Rukhlenko ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.8b02867 • Publication Date (Web): 29 May 2018 Downloaded from http://pubs.acs.org on May 29, 2018

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Optically Active Semiconductor Nanosprings for Tunable Chiral Nanophotonics Anvar S. Baimuratov,† Tatiana P. Pereziabova,† Mikhail Yu. Leonov,† Weiren Zhu,‡ Alexander V. Baranov,† Anatoly V. Fedorov,† Yurii K. Gun’ko,¶ and Ivan D. Rukhlenko∗,† †Information Optical Technology Centre, ITMO University, Saint Petersburg 197101, Russia ‡Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China ¶School of Chemistry and CRANN Institute, Trinity College Dublin, Dublin 2, Ireland E-mail: [email protected]

Abstract The search for the optimal geometry of optically active semiconductor nanostructures is making steady progress and has far-reaching benefits. Yet the helical spring-like shape, which is very likely to provide a highly dissymmetric optical response, remains somewhat under-studied theoretically. Here we comprehensively analyse the optical activity of semiconductor nanosprings using a fully quantum-mechanical model of their electronic subsystem and taking into account the anisotropy of their interaction with light. We show that the circular dichroism of semiconductor nanosprings can exceed that of ordinary semiconductor nanocrystals by a factor of 100 and be comparable to the circular dichroism of metallic nanosprings. It is also demonstrated that nanosprings can feature a total dissymmetry of optical response for certain ratios between their

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length and coil height. The magnitude and sign of the circular dichroism signal can be controlled by stretching or compressing the nanosprings, which makes them a promising material base for optomechanical sensors, polarization controllers, and other types of optically active nanophotonic devices.

Keywords: absorption, circular dichroism; optomechanics; helical symmetry; tunable optical properties

The strength of optical activity of chiral semiconductor nanostructures is determined by the type and location of their dissymmetry centers as well as by the ratio of the nanostructure size and the excitation wavelength. 1 The strongest optical activity — close to that of chiral plasmonic complexes — is exhibited by relatively large chiral quantum-dot molecules 2,3 and supercrystals made of such molecules. 4–6 Individual nanocrystals possess much weaker optical activity, which can still significantly exceed the activity of small organic molecules. 7 The optical activity of semiconductor nanocrystals can be induced by chiral ligands attached to the nanocrystal surface, 8–11 various kinds of bulk defects, including dislocations, disclinations, and impurities, 12–14 as well as by the nanocrystal shape. 15 Of particular interest from the viewpoint of maximising the strength of chiroptical response are nanostructures of helical symmetry, 16 which is inherent to many kinds of optically active organic molecules, including DNA, proteins, peptides, amino acids, and sugars. 17 Examples of such nanostructures include ZnO and CdS nanohelices, 18,19 carbon-based nanocoils, 20 SiO2 nanosprings, 21 and nanosprings made of II-VI and III-V semiconductors. 22 These nanostructures are routinely fabricated these days using chemical vapor deposition, 23–25 glancing angle deposition, 26 asymmetric active surface growth, 27 and various template methods. 28 Owing to their useful electronic, optical, and mechanical properties, helical inorganic nanostructures are showing promise for use in a wide variety of applications. 1 They have already proven themselves useful as components for microelectromechanical and nanoelec2


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tromechanical systems, 29 nanoscale sensors, 30 and hydrogen-storage devices. 31 In particular, semiconductor nanosprings can serve as a basis of an efficient photosensor platform due to their higher optical absorption as compared to ordinary nanowires. 32 Because semiconductor nanosprings combine both helical shape and chiral crystal lattice, one may also expect that the dissymmetry of their interaction with different polarisations of circularly polarised light can be especially strong. Despite a great deal of experimental research directed at understanding the optics of semiconductor nanosprings, a complete theory of these three-dimensional nanostructures is still needed for quantifying their chiral optical properties. The existing theoretical models do not allow one to perform such a quantification, because they simplistically treat real nanosprings as one-dimensional 33–36 or two-dimensional nanostructures. 37,38 This work presents a first quantum-mechanical theory of optical activity of semiconductor nanosprings, which takes into account the true three-dimensional nature of the spatial confinement inside them and the essential anisotropy of the nanospring interaction with light. The developed theory is then used to calculate the absorption and circular dichroism spectra of the nanosprings and to analyse how their chiroptical response depends on the geometric and material parameters of the system. We also demonstrate how the mechanical deformation of semiconductor nanosprings enables an efficient dynamic control over their optical properties.

Results and discussion In our model we consider a cylindrical semiconductor nanospring of length L and uniform coil height b centered along the z axis. Let the coil height be positive for right-handed nanosprings and negative for left-handed ones. We specify the nanospring geometry by the following six boundaries: planes z = 0 and z = L, concentric cylinders of radii ρ0 and ρ1 > ρ0 , and helicoids z = βϕ and z = β(ϕ − ϕ0 ), where β = b/(2π) and the angular confinement parameter ϕ0 represents the central angle of the sector that is cut off from the

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z

a

b

c

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d

e

�0 �0

�1

L b

|� 111|

|� 211|

|� 121|

|� 112|

Figure 1: (a) Two-coil semiconductor nanospring and (b)–(e) chiral isosurfaces of the envelope wave functions of its ground state ψ111 and three excited states ψ211 , ψ121 , and ψ112 . cylinders by the helicoids. Thus defined, our nanospring turns into an achiral nanotube for ϕ0 = 2π and into a chiral nanoribbon for ϕ0 π or ρ0 ≈ ρ1 . The geometric parameters of the system are illustrated by the example of the right-handed nanospring with two coils shown in Fig. 1(a). The motion of charge carriers confined to the nanospring is roughly bounded by the nanospring surface. In order to analytically calculate the envelope wave functions ψ and energy spectrum E of the carriers, we assume that the nanospring surface is impenetrable for both electrons and holes, and employ the effective mass approximation and the two band model of semiconductor. 15 The description of the confined motion should take into account the chirality of the nanospring’s crystal structure resulting from the internal strain of the nanospring. This can be done using the geometric theory of defects via the replacement of the Laplace operator with the Laplace–Beltrami operator. 39 Given the helical symmetry of the nanospring, this amounts to the replacement of the z derivative by ∂/∂z+(1/β)∂/∂ϕ. 40,41 The resulting Schrödinger equation reads 2 ∂ 1 ∂2 ∂ 1 ∂ ~2 1 ∂ ρ + 2 2 + + ψ = (E − V )ψ, − 2m ρ ∂ρ ∂ρ ρ ∂ ϕ ∂z β ∂ϕ

(1)

where m is the scalar effective mass of electrons (or holes) and the confining potential V is zero inside the nanospring and infinite outside of it. By noticing that our nanospring turns into an achiral nanoscroll in the limiting case of

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β → ∞, we can write the solution of Eq. (1) by straightforwardly generalising the analogous solution available for semiconductor nanoscrolls. 42 The resulting states of the nanospring are characterised by three nonzero integers n, l, and p and are given b

ψnlp =

2 √ rnl (ρ/ρ1 ) sin[λ(ϕ − z/β)] sin kp z, ρ1 ϕ0 L 2 ~2 ζnl 2 Enlp = + kp , 2m ρ1

(2) (3)

where rnl (x) = anl [Jλ (ζnl x) + bnl Yλ (ζnl x)], Jλ (x) and Yλ (x) are the Bessel functions of the first and the second kinds, λ = πl/ϕ0 , ζnl is the nth root of equation Jλ (ζnl )Yλ (ζnl ρ0 /ρ1 ) = Jλ (ζnl ρ0 /ρ1 )Yλ (ζnl ), kp = πp/L, and coefficients anl and bnl are given in Supporting Information. It should be noted that Eq. (3) also gives the energy spectrum of a straight nanowire with an approximately square cross section of (ρ1 − ρ0 )ρ1 ϕ0 . The wave functions of the first four quantum states are shown in Figs. 1(b)–1(e). The lack of planes and center of symmetry makes semiconductor nanosprings optically active. Without loss of generality, we quantify the optical activity of nanosprings by assuming that their conduction band contains one extra electron, which comes from a donor-type impurity or is injected into the system. The interaction of this electron with a circularly polarised plane wave of frequency ω and polarization vector eχ (χ = L or R) is described by the Hamiltonian

Hχ = −

e A(ω)eik·r (eχ · p), mc

(4)

where c and −e are the speed of light in vacuum and the charge of free electron, A(ω) is the √ amplitude of vector potential, k = εk u, ε is high-frequency permittivity of the nanospring, k = ω/c, u is the unit vector of the propagation direction, and r and p = −i~∇ are the radius vector and momentum operators. Following Rosenfeld’s theory of optical activity in molecular systems, 43,44 we assume that

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the intraband matrix elements of the radius vector are much smaller than the characteristic √ length scale 1/( εk), and approximate the exponential function in the Hamiltonian by the first two terms in its Taylor series expansion as eik·r ≈ 1+i(k · r) (the entire exponential must be retained for relatively long nanosprings). 12 We also restrict our consideration to intraband transitions and excitation energies smaller than the bandgap Eg of bulk semiconductor. With this restriction, the adopted approximation is quite accurate for 50-nm-long CdSe nanosprings with Eg = 1.74 eV and ε = 6.2, for example. A semiconductor nanospring excited by circularly polarised light transfers from its initial state |ii = |nlpi to a higher-energy final state |f i = |n0 l0 p0 i. Fermi’s golden rule and Eq. (4) yield the following sum (+) and difference (−) of transition probabilities corresponding to the left-circularly polarised and right-circularly polarised light: 44 (+)

wi→f (ω) =

2 2 i Ch |pf i |2 − |pf i · u|2 + εk 2 (p ⊗ r)f i · u − u · (p ⊗ r)f i · u δ(∆Ef i ) (5) 2k

and √ (−) wi→f (ω) = − C ε u · pif × (p ⊗ r)f i · u δ(∆Ef i ),

(6)

where C = [4πe/(mc)]2 I/(~ω), I = ω 2 |A(ω)|2 /(2πc) is the excitation light intensity, pf i = hf |p|ii, (p ⊗ r)f i = hf |p ⊗ r|ii, ⊗ denotes the tensor product of vectors, δ(x) is the Dirac delta function, and ∆Ef i = ~ω − Ef + Ei . Using cyclic covariant coordinates (+1, 0, −1) we

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can write   ~  p = −√  2  

iφ

 − sin θ e √ 1  u= √  2 cos θ  2 sin θ e−iφ

∂ eiϕ ρ1 ∂ϕ

√

−

∂ i ∂ρ

∂ i 2 ∂z

e−iϕ    ,  

1 ∂ ρ ∂ϕ

∂ + i ∂ρ



  ,   

(7)

iϕ

 −ρe √ 1  r= √  2z  2 ρe−iϕ

   ,  

(8)

where θ and φ are the spherical coordinates of u. Equations (5) and (6) show that the interaction of nanosprings with circularly polarised light is essentially anisotropic. Evaluation of the matrix elements and tensor products in these (±)

equations leads to the analytic expressions for probabilities wi→f (ω). These expressions are given in Supporting Information. The developed theory allows one to calculate the absorption and CD signal of semiconductor nanosprings. When the spacing between the electronic energy levels is comparable to the thermal energy kB T , both signals depend on the occupation probabilities of the levels. By considering that the nanospring length often significantly exceeds its cross-sectional dimensions, we assume that the energy spacing Enlp0 − Enlp is comparable to the thermal energy for p ∼ p0 and that En0 l0 p0 − Enlp kB T for nl 6= n0 l0 . Because there is only one extra electron in the system, the occupation probability sp of level p in any subband is given by the Boltzmann distribution

s−1 p

=

X

exp

j=1,2,...

~2 π 2 p2 − j 2 . 2mL2 kB T

(9)

This probability is seen to critically depend on the electron confinement strength along the z direction determined by the nanospring length L. In the case of relatively strong confinement p with L ~π/ 2mkB T /3, only the lowest-energy state with p = 1 is effectively occupied.

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Consider an ensemble of noninteracting and equally oriented nanosprings of concentration N , and suppose that the conduction electron of each nanospring initially occupies the subband with quantum numbers nl. Then the absorption and CD spectra of the ensemble are obtained by summing over the probabilities of transitions from this subband to all the possible final states |f i of other subbands. The resulting spectra are given by

Anl (ω) =

N ~ω X (+) sp wnlp→f (ω), 2n0 I f,p

CDnl (ω) =

N ~ω X (−) sp wnlp→f (ω), n0 I f,p

(10) (11)

where n0 is the refractive index of the medium through which the light propagates and (±)

delta functions δ(E) in wnlp→f (ω) are replaced by Lorentzians γ/[π(E 2 + γ 2 )] to allow for the natural linewidths of the spectral lines. These expressions become less accurate as the interaction between the nanosprings becomes significant and enhances their optical activity. 45 The CD signal of real samples is often measured by its ellipticity, which scales in proportion to the sample thickness h and is given by (see Supporting Information)

ϑnl (ω) ≈

45h CDnl (ω) (deg). π

(12)

Figure 2 illustrates the anisotropy of absorption and circular dichroism upon intraband transitions from the ground subband (n = l = 1) of CdSe nanosprings at room temperature. According to the selection rules (see Supporting Information), the lowest-energy absorption peak and the first two CD peaks correspond to the electronic transitions of the form |11pi → |12p0 i. The negative CD peak, centered at 260 meV, coincides with the absorption peak originating from the dipole-allowed transitions, whereas the positive CD peak at 310 meV corresponds to weaker absorption upon the dipole-forbidden transitions. The dissymmetry factors of these negative and positive CD peaks are estimated to be −0.02 and 0.04, respectively, in the agreement with the general rule that the dissymmetry factors of

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a

b

c

d

e

ϕ

f

z

z

θ

Figure 2: Absorption spectra (left) and CD spectra (right) of CdSe nanosprings as functions of the propagation direction of the excitation light for n = l = 1, [(a) and (b)] φ = π/3, and [(c) and (d)] θ = π/3. Panels (e) and (f) are the spherical plots of A11 (θ, φ) and |CD11 (θ, φ)| for ~ω = 260 meV. Simulation parameters are: ρ0 = 10 nm, ρ1 = 13 nm, L = 50 nm, ϕ0 = π/6, b = 33 nm, m = 0.12 m0 (m0 is the free-electron mass), ε = 6.2, N = 1014 cm−3 , n0 = 2.5, T = 300 K, and γ = 20 meV. the electric dipole transitions are larger than those of the magnetic dipole ones. 41 The strongest absorption occurs when the nanosprings are excited along their axis, and the maximal CD signal is observed when k is slightly inclined with respect to this axis due to the finite nanospring length. This is clearly seen from the spherical plots of functions A11 (θ, φ) and |CD11 (θ, φ)| shown in Figs. 2(e) and 2(f) for ~ω = 260 meV. The absorption plot has a relatively broad waist perpendicular to the z axis whereas the CD signal is strongly anisotropic and represents an approximately 14◦ -inclined dumbbell. The spherical plot of the CD signal at 310 meV is a similar dumbbell rotated by π around the z axis. Due to the helical symmetry of semiconductor nanosprings, the absolute strength of their

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Dissymmetry factor

a 0.2

�0 �0 �0 �0

b

= 7 nm; �1 = 13 nm

0.3

= 10 nm; �1 = 13 nm = 10 nm; �1 = 16 nm

0.2

= 13 nm; �1 = 16 nm

0.1

0

-0.1

-0.1 0.3

0.2 0.1

c

L = 50 nm L = 25 nm L = 10 nm L = 5 nm

0.2 0.1

�0 = �/6 �0 = �/3 �0 = �/2 �0 = �

0.1

0

d Dissymmetry factor

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0.6

0.9

1.2

1.5

-0.1 0.3

e

b = 23 nm b = 33 nm b = 43 nm b = 53 nm

0

0.2 0.1

0.6

0.9

1.2

0.3

1.5

f

T = 300 K T = 100 K T = 10 K T=5K

0.3 0.2

0.6

0.9

1.2

1.5

��= 10 meV ��= 20 meV ��= 30 meV � = 40 meV

0.1 0

0

-0.1

-0.1 0.3

0.6 0.9 1.2 Photon energy (eV)

1.5

0 -0.1 0.3


1.5

0.3


1.5

Figure 3: Dissymmetry factor gnl (ω) for various geometric and material parameters of CdSe nanosprings excited along the z axis (θ = 0). In all cases, red spectrum corresponds to the simulation parameters in Fig. 2. optical activity is extremely high. For example, the ellipticity of the CD peak at 260 meV is about 1 deg for h = 1 cm. The optical activities of such magnitude are typical for metallic nanosrings 46 and have not been reported in the available literature for semiconductor nanocrystals until now to the best of our knowledge. For example, the CD signal produced by semiconductor nanosprings is about 100 times larger than the CD signal of tapered semiconductor nanoscrolls of the same concentration. 41 Similar to the dissymmetry factor of optical transitions, we introduce the spectrum of dissymmetry factor as the ratio of the CD and absorption spectra, gnl (ω) = CDnl (ω)/Anl (ω). 44 Figure 3 shows how this spectrum modifies with various geometric and material parameters of CdSe nanosprings excited along the z axis. The largest shifts in the dissymmetry factor spectrum are seen to occur on panel (a) upon the change of the radial confinement. This feature can be used to fine-tune the spectral positions of the nanospring enantioselectivity peaks as desired for applications. Fabrication of such a nanospring would be quite challenging due to the need of achieving radial confinement in the order of a few nanometers. At

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2 Dissymmetry factor

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p' = 1 p' = 2

1 0

p' = 3 p' = 4

-1 -2

15

20

25 30 Coil height (nm)

35

40

Figure 4: Dissymmetry factors of four intraband transitions |111i → |12p0 i (p0 = 1, 2, 3, 4) as functions of coil height for L = 50 nm. The nanospring is excited along the z axis (θ = 0). Simulation parameters are the same as in Fig. 2. this moment it is possible to achieve radial confinements in the order of tens of nanometers. 19 Much smaller, yet still pronounced blue shifts are obtained on panel (b) upon the strengthening of the confinement in the z direction, which is accompanied by the growth of the peak dissymmetry factors. The dissymmetry factor peaks also decrease with both the angle confinement parameter ϕ0 and the coil height b, as evidenced by panels (c) and (d). The impact of temperature on the optical activity of semiconductor nanosprings is twofold. First, its increase slightly blueshifts the spectrum of the dissymmetry factor [see panel (e)] due to the rise in the probabilities of the electron to occupy higher-energy states. Second, the natural broadening of spectral lines with temperature weakens the peaks of gnl (ω) [see panel (f)] due to the destructive interference of the closely spaced allowed and forbidden transitions with the opposite CD signs. These effects of temperature become more pronounced for nanosprings with two or more electrons, whose energy distribution is described by Fermi–Dirac statistics. The maximal dissymmetry of the nanospring interaction with two circular polarisations can be estimated by calculating the dissymmetry factor of individual transitions, gif = (−)

(+)

2wi→f /wi→f . 47 It turns out that for any intraband transition we can choose nanospring dimensions such that gif = ±2, which implies that semiconductor nanosprings can fully absorb light of one circular polarisation and not interact with the light of the other. Figure 4

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shows how gif varies with coil height for four transitions from the lowest-energy state |ii = |111i of a 50-nm-long CdSe nanospring. The dissymmetry factor is seen to change sign for b = L/2, 2L/5, L/3, and 2L/7, reaching its extremums for coil heights slightly above or below these values. In the case of transition to state |f i = |121i for example, gif = ±2 for b = 25 ± 0.4 nm and b = 16.67 ± 0.17 nm. Note that the change of the CD sign in this case is not accompanied by a change of the nanospring handedness, showing that semiconductor nanosprings can strongly interact with both circular polarizations regardless of their twist. Furthermore, by independently varying the three dimensions of the nanospring, one can make its interaction with circular polarised light totally dissymmetric at any frequency of interest. In practice, this would require using cryogenic temperatures to prevent the overlapping and partial cancellation of the positive and negative CD peaks. It should also be noted that the total dissymmetry of optical response has been reported so far only for infinitely long helical quantum-dot supercrystals 6 and for coupled semiconductor nanocuboids with extremely finely tuned dimensions. 3 Chiral plasmonic nanostructures, on the other hand, have demonstrated much lower dissymmetry factors, with a maximal value of about 0.4 reported upon transmission for micrometer-size plasmonic nanosprings. 48 The pronounced dependence of the optical properties of nanosprings on the nanospring geometry allows one to control these properties dynamically. This can be done, in particular, by compressing or stretching a polymer membrane with embedded nanosprings. The deformation of the membrane is transferred to the nanosprings, altering their energy spectrum and absorption efficiency. To demonstrate this possibility, consider identical and equally oriented nanosprings excited along their axis. A uniform deformation of the nanosprings in the z direction can be described via the replacement of their three geometric parameters as ϕ0 → αϕ0 , L → L/α, and β → β/α, where α > 1 corresponds to compression and α < 1 represents stretching. Figures 5(a) and 5(b) show how the intensities of the strongest absorption and CD peaks (centered at 260 meV) change with the nanospring compression. The CD signal increases from −0.08 cm−1 for α = 1 to nearly zero for α = 1.06 and reaches 0.04 cm−1

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a

b

CD < 0 CD = 0 CD > 0

c

d

e

Figure 5: (a) Compression or stretching of CdSe nanosprings modifies their (b) absorption and CD peaks at 260 meV as well as the entire (c) absorption, (d) CD, and (e) dissymmetry factor spectra. It is assumed that θ = 0 and the rest of simulation parameters are the same as in Fig. 2 for α = 1.13 while the respective dissymmetry factor increases from −0.02 to 0.03. The change in the CD sign is caused by the red shift of the negative and positive CD peaks due to the weakening of the angular confinement. The extents of the deformation-induced modifications of the absorption, CD, and dissymmetry factor spectra illustrated by Figs. 5(c) and 5(d) show that semiconductor nanosprings can serve as a versatile material base for optically active nanophotonic devices such as optomechanical sensors and polarization controllers.

Conclusions In summary, we have comprehensively analysed the chiral optical properties of semiconductor nanosprings with impenetrable boundaries. Using the exact wave functions and energies

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of electrons confined to CdSe nanosprings for example, the CD signal, its anisotropy, and its dependence on the nanospring parameters were discussed in details. We found that the CD signal of semiconductor nanosprings is strongly anisotropic, characterised by a dumbbellshape intensity distribution, and can exceed the recently reported CD signals of chiral semiconductor nanocrystals by a factor of 100. It was also shown that nanosprings with certain ratios of length and coil height can exhibit a total dissymmetry of their optical response. The performed analysis finally revealed that stretching or compression of semiconductor nanosprings enables an efficient dynamic control over the magnitude and sign of their CD signal. This concept may prove useful for the design of superior nanophotonic devices for chiral optomechanics, biochemistry, and biomedical applications.

Methods All the analytical calculations given in Supporting Information were performed using Wolfram Mathematica 11 and numerical simulations were carried out using MATLAB R2017a.

Supporting Information Available Supporting Information details calculations of wave functions and energy spectrum of semiconductor nanosprings; absorption rates of circularly polarized light; matrix elements of intraband transitions; occupation probabilities of quantum states at finite temperatures; absorption and CD spectra; selection rules of intraband transitions; and the ellipticity of CD signal.

Acknowledgement We thank the Ministry of Education and Science of the Russian Federation for its Project 14.Y26.31.0028, Scholarships SP-2066.2016.1 and SP-1975.2016.1, and Grant of the Pres14


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ident of the Russian Federation for Young Scientists MD-1294.2017.2. W. Zhu gratefully acknowledges National Natural Science Foundation of China (61701303), Natural Science Foundation of Shanghai (17ZR1414300), and Shanghai Pujiang Program (17PJ1404100).

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Graphical TOC Entry

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0 Absorption CD 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Compression/streching ratio

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Optically Active Semiconductor Nanosprings for Tunable Chiral

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