Abstract

Chiral molecules made of coupled achiral semiconductor nanocrystals, also known as quantum dots, show great promise for photonic applications owing to their prospective uses as configurable building blocks for optically active structures, materials, and devices. Here we present a simple model of optically active quantum-dot molecules, in which each of the quantum dots is assigned a dipole moment associated with the fundamental interband transition between the size-quantized states of its confined charge carriers. This model is used to analytically calculate the rotatory strengths of optical transitions occurring upon the excitation of chiral dimers, trimers, and tetramers of general configurations. The rotatory strengths of such quantum-dot molecules are found to exceed the typical rotatory strengths of chiral molecules by five to six orders of magnitude. We also study how the optical activity of quantum-dot molecules shows up in their circular dichroism spectra when the energy gap between the molecular states is much smaller than the states’ lifetime, and maximize the strengths of the circular dichroism peaks by optimizing orientations of the quantum dots in the molecules. Our analytical results provide clear design guidelines for quantum-dot molecules and can prove useful in engineering optically active quantum-dot supercrystals and photonic devices.

© 2017 Optical Society of America

1. Introduction

As the name suggests, quantum-dot molecules are the quantum-mechanical systems which are made from coupled semiconductor nanocrystals (quantum dots) and have delocalized electronic states and molecular-like wave functions [1]. Much like the naturally occurring molecules are the elementary building blocks of molecular crystals, quantum-dot molecules can serve as the building blocks of higher-order structures — quantum-dot supercrystals [2]. Owing to their unique and readjustable optical properties, which significantly differ from the properties of noninteracting quantum dots and can be individually tuned over wide ranges by changing the number, relative positions, sizes, and shapes of individual nanocrystals, quantum-dot molecules is a very promising material base for the next-generation quantum information technologies and devices. The last five years have already witnessed various experimental realizations of such molecules [1] and their multiple uses in the implementation of solar cells [3], qubits [4,5], logic gates [6], and quantum memories [7].

The optical activity of chiral molecules is naturally weak due to the smallness of molecules compared to the excitation wavelength and the relatively low density of the constituting atoms [8, 9]. While semiconductor nanocrystals are also much smaller than the optical wavelength, they can be anticipated to exhibit a much stronger optical activity due to the significantly higher density of atoms. A great deal of research efforts have therefore been recently focussed on studying the chiroptical response of individual chiral nanocrystals [10–13], achiral quantum-dot molecules [14, 15], and quantum-dot supercrystals [16, 17]. In particular, it was theoretically shown that otherwise optically inactive semiconductor nanocrystals exhibit strong optical activity in the presence of screw dislocations [18], asymmetrically positioned inside them ionic impurities [19], and chiral surface defects [20]. This activity was found to be significantly stronger than that of chiral organic molecules whereas the asymmetry in the interaction of these chiral nanocrystals with left- and right-handed light was shown to be comparable to that of chiral plasmonic complexes [21–28]. It was also shown that by assembling achiral quantum dots into helix-like structures much longer than the excitation wavelength, one can make them fully absorb light of one circular polarization and transmit the light of the other [29].

This paper presents the first theory of chiral quantum-dot molecules, which allows one to analytically treat and optimize the optical activity of basic configurations of such coupled quantum systems. We begin with a formulation of the general theoretical model of artificial molecules made from an arbitrary number of sufficiently strongly coupled semiconductor quantum dots. Each of the quantum dots is assumed to have a dipole moment associated with the fundamental interband transition between the size-quantized states of its confined charge carriers. This assumption amounts to considering quantum dots as elongated nanorods with their largest dimension along the growth direction of the crystal structure. We use the developed model to analytically calculate the wave functions, energy spectrum, rotatory strengths, dipole absorption rates, and dissymmetry factors of quantum-dot dimers, trimers, and tetramers. The obtained expressions are then thoroughly analyzed to find the optimal configurations of the quantum-dot molecules featuring the strongest chiroptical response.

2. Theoretical model

Consider a chiral molecule composed of N arbitrarily oriented semiconductor nanocrystals (quantum dots) with coordinates r1, r2, . . . , rN. Let all the quantum dots be identical in terms of size, shape, and chemical composition. Then the wave function of the first excited electronic state ψn(r) in the nth quantum dot can be represented in the form ψn(r) = ψ(rrn). This state is assumed to be strongly localized inside the quantum dots, so that the wave functions of different nanocrystals do not overlap and are normalized according to the condition

Vnψn(r)ψm*(r)dr=δnm,
where the integration is evaluated over the volume Vn of the nth quantum dot and δnm is the Kronecker delta.

If the interaction between the quantum dots is negligibly small, then the first excited state of the quantum-dot molecule is N-fold degenerate and has energy E of state ψ. The interdot coupling partially or fully lifts this degeneracy [30], leading to the formation of delocalized molecular states Ψ1, Ψ2, . . . , ΨN of energies E1, E2, . . . , EN. These states obey the stationary Schrödinger equation ĤΨμ = EμΨμ and can be experimentally studied by analysing the absorption and luminescence spectra of the molecule. By denoting the matrix element of the interaction between dots n and m as Vnm, we can write the Hamiltonian of the quantum-dot molecule in the form

H=(EV12V1NV21EV2NVN1VN2E).
Each of the molecular states is the linear superposition of the quantum-dot states,
Ψμ=cμ1ψ1+cμ2ψ2++cμNψNcμnψn.
Hereinafter we adopt the Einstein summation convention to achieve notational brevity. The energies Eμ of the molecular states and coefficients cμn — normalized according to the condition |cμ1|2 + |cμ2|2 + . . . + |cμN|2 = 1 — are the solution to the eigensystem problem (HmnδmnEμ)cμn = 0.

The rotatory strength of the interband transition that creates an electron in state Ψμ is given by the Rosenfeld’s formula [31,32]

Rμ=e2ε2cEgm2Re(p0μ[r×p]μ0),
where −e and m are the charge and mass of a free electron, Eg is the effective bandgap, which is the sum of the bandgap energy of bulk semiconductor and the confinement energy, c is the speed of light in a vacuum, and ε is the effective high-frequency permittivity of the quantum dot. The matrix elements in this formula can be expressed using Eq. (3) through the matrix elements pn and rn of the electric dipole moment and radius vector as
pμ0=Ψμ|p^|0=cμn*ψn|p^|0=cμn*pn
and
[r×p]μ0=Ψμ|r^×p^|0=Ψμ|r^×pn|ψn=cμm*ψm|r^×pn|ψn=cμn*[rn×pn],
where we have taken into account that |ψn〉〈ψn||0〉 = pn|ψn〉 and defined rn as the average value of the radius vector in state ψ by assuming that 〈ψm||ψn〉 = rnδmn. By treating all the quantum dots in a molecule as point dipoles dn (|d1| = |d2| = . . . = |dN| = d), which are related to the interband matrix elements of the dipole moment as pn = −i[mEg/()]dn, we obtain the following expression for the rotatory strength:
Rμ=σRe(cμncμm*)(dn[rm×dm]),
where σ=εEg/(2c).

One can see that regardless of the nature of the interaction between the quantum dots, the rotatory strength vanishes if all the dipoles are parallel to each other. The fact that this strength also vanishes in the hypothetical situation where the displacement vectors of the quantum dots are all alike (r1 = r2 = . . . = rN) implies that the rotatory strength is independent of the origin of coordinates. It should also be noted that the rotatory strength Rμ is proportional to the total optical helicity emitted by the quantum dot molecule (see Ref. [33] and, more specifically, Eq. (5) from Ref. [34]).

The full dipole absorption rate upon the excitation of state Ψμ is given by the sum [10]

Dμ=|cμndn|2+σ2|cμn[rn×dn]|2.
This expression allows one to calculate the dissymmetry factor (g-factor) of the interband transition to state μ using the definition gμ = 4Rμ/Dμ. This dissymmetry factor is proportional to the normalized chirality of the dipole emitters constituting the molecule (see, e.g., Eq. (4) from Ref. [35]).

Finally, in order to complete the above model of optical activity, we approximate the coupling between the quantum dots inside molecules by the dipole–dipole interaction potential

Vnm=1εrnm5[rnm2(dndm)3(rnmdn)(rnmdn)],
where rnm = rnrm.

3. Quantum-dot dimer

A quantum-dot molecule exhibits optical activity if it lacks a center or plane of symmetry. We now apply the developed general formalism to three kinds of quantum-dot molecules with the aim of optimizing their structure and achieving the strongest manifestation of optical activity in the circular dichroism (CD) spectra.

We begin by considering the simplest molecule consisting of two identical quantum dots, each of which has its own direction of the dipole moment. The dipole moment of the jth quantum dot (j = 1 or 2) is described by a pair of angles (ϑj, φj) shown in Fig. 1(a). The interaction between the dipoles lifts the double degeneracy of the molecular states, producing a pair of optically active states of energies E1,2 = E ± δE, where δE is given by Eq. (19). The energy gap between the dimer states scales like ∝ d2/(εa3) whereas the rotatory strengths of the transitions to these states scale like ∝ σad2, where a is the distance between the quantum dots. These two scaling factors are characteristic of quantum-dot molecules with the dipole–dipole coupling. The angular dependencies of the energy gap and rotatory strengths are of the forms (see Appendix A)

δEsinϑ1sinϑ2cos(φ2φ1)2cosϑ1cosϑ2,
R1,2sinϑ1sinϑ2sin(φ1φ2).
An interesting peculiarity of this result is that the peaks of δE, attained for ϑ1 and ϑ2 equal to 0 or π, correspond to the zero of Rj. This feature simply reflects the fact that the maximal energy splitting is attained for the most strongly interacting parallel dipoles constituting an essentially achiral molecule. It also suggests the need of the tradeoff between δE and Rj for the achievement of the most pronounced lines in the CD spectra.

 

Fig. 1 Quantum-dot (a) dimer, (b) trimer, and (c) tetramer composed of equal but differently oriented semiconductor quantum dots. The molecules become optically active for certain orientations of dipole moments dj of the fundamental interband transitons inside the quantum dots; a is the distance between the quantum dots in the dimer, trimer, and in the base of the tetramer; b is the height of the tetramer.

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Suppose that the lineshapes of the two transitions can be approximated by Lorentzians Γ(Ej, ω) = (γ/π)/[(ħωEj)2 + γ2] of the full width at half maximum (FWHM) 2γ. Then the CD spectrum of the quantum-dot dimer has the form

CD(R1,δE,γ,ω)R1Γ(E1,ω)+R2Γ(E2,ω)γ(ωE)R1δE[(ωE+δE)2+γ2][(ωEδE)2+γ2].
This expression shows that the optimal values of ϑ1, ϑ2, and δφ = φ1φ2 generally depend on the width of the spectral lines. These values are easy to find for relatively broad spectral lines (with γδE), which peak at ω=E±γ/3 and whose intensities scale like ∝ R1δE. Some algebra shows (see Appendix A) that the strongest CD peaks are manifested by dimers with
δφ=±2π3,ϑ1=ϑ2=12arccos(13)54.7°.
These angles result in δE = −η and R1=ρ/3, where η = d2/(εa3) and ρ = σ (a/2) d2. There are also four optimal dimers with 3/2 times weaker CD peaks. Their parameters — ϑ1 = ϑ2 = π/2 and δφ = ±π/4 or ±3π/4 — result in R1=ρ/2 and δE=η/2 or η/2. It will be shown below that dimers with different absolute values of δφ possess different dissymmetry factors.

Figures 2(a)–2(c) show one kind of enantiomers (with negative δφ) of the three optimal quantum-dot dimers. While dimer (a) corresponds to the absolute maximum of the CD peaks, dimers (b) and (c) are the saddle-point configurations. Therefore, there is a continuum of optimal dimer configurations, with their CD-peak intensities spanning between those of dimers (a) and (c). These configurations are schematically shown by the shaded regions in Fig. 2(d).

 

Fig. 2 Optimal configurations of quantum-dot dimers exhibiting the strongest peaks in the CD spectrum for γδE. The dipoles of the upper quantum dots are all in the yz plane. The intensities of peaks in the CD spectrum of dimer (a) exceed the intensities of peaks in the spectra of dimers (b) and (c) by a factor of 3/2. Panel (d) shows the transformation between the optimal configurations (c) and (a), which leads to a continuum of intermediate optimal orientations of the quantum-dot dipoles.

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As we have seen, the linear growth of the rotatory strength with the distance between the quantum dots due to the increase of the angular momentum is accompanied by the much faster reduction of the energy gap between the two molecular states (like ∝ a−3) due to the weakening of the dipole–dipole coupling. This eventually reduces the intensity of the CD signal. The dependence of the dissymmetry factors on a is somewhat more intricate. For ϑ1 = ϑ2 these factors are given by the expressions (see Appendix A)

g1=4χsinδφ2(cosδφ1)χ2,g2=4χsinδφ2(cosδφ1)+4csc2ϑ1+χ2,
where χ = is a dimensionless coefficient, which determines the relative strengths of the electric-dipole and magnetic-dipole absorption rates.

The absolute values of both dissymmetry factors are seen to grow like ∝ a for a ≪ 1/σ when the electric-dipole absorption is stronger than the magnetic-dipole absorption, and to decrease like ∝ 1/a when a becomes large and the magnetic-dipole absorption starts dominating the electric-dipole one. The maximal values g1 = − χ1 cot(δφ/2) and g2 = (2/χ2) sin δφ are attained for χ1=2(1cosδφ) and χ2=4csc2ϑ12(1cosδφ). For example, g1=1/20.71 and g2 = ±1 are the maximal dissymmetry factors for the dimers whose parameters are given in Eq. (13). The analogous values for the dimers shown in Figs. 2(b) and 2(c) are g1=1/g2=1/221.31 and g1=1/g2=1/2+20.54 respectively. These values are about 103 − 104 times larger than the typical g-factors of chiral organic molecules and can exceed the dissymmetry factors of intrinsically chiral nanocrystals by more than an order of magnitude [11,36,37].

It is instructive to estimate the typical values of the scaling factors σ, η, and ρ. By assuming the following material parameters: ε = 5, Eg = 2 eV, d = 200 D, and a = 7 nm, we get σ−1 ≈ 75 nm, η ≈ 14.6 meV, and ρ ≈ 1.88 × 10−33 erg×cm3. Hence, the rotatory strengths of quantum-dot molecules can be up to 104 times larger than the rotatory strengths of chiral semiconductor nanocrystals doped with impurity ions [19], and can exceed the rotatory strengths of chiral molecules by five to six orders of magnitude [38–40]. The obtained estimate also justifies the approximation of broad spectral lines (γδE), which was employed to study the optimal dimer configurations. Indeed, the energy parameter η is much smaller than the typical linewidths (∼ 100 meV) of interband transitions in semiconductor quantum dots at room temperature [41,42].

4. Quantum-dot trimer

Consider next three quantum dots located in the vertices of an equilateral triangle shown in Fig. 1(b). Let the direction of the dipole moment of the first quantum dot be set by the polar angle ξ and the azimuthal angle φ whereas the dipole moments of the second and third quantum dots have the same polar angle ϑ and azimuths φ + 2π/3 and φ − 2π/3 respectively. This molecule has three generally nondegenerate and optically active states (see Appendix B) of energies E1 = EB and E2,3 = E + δE, where δE±=B/2±2A2+(B/2)2 and coefficients A and B given in Eq. (33) are the functions of ξ, ϑ, and φ. The rotatory strengths of all the states vanish in essentially achiral molecules with φ = π/2 or 3π/2.

Trimers with ξ = −ϑ are of particular interest, because the sign of their rotatory strength R1 is different to the signs of the other two rotatory strengths regardless of the specific directions of the quantum-dot dipoles. Unless φ is specifically chosen such that δE = −B, there can be up to three peaks in the CD spectrum of such trimers, the strongest of which is often produced by the first molecular state. If this peak is much narrower than the energy spacing between states Ψ1 and Ψ2, which occurs for γB + δE, then its intensity scales in direct proportion to the rotatory strength R1 ∝ sin 2ϑ cos φ. In this case, the optimal trimers are described by parameters ϑ = π/4 or 3π/4 and φ = 0 or π. One of the optimal trimer configurations is shown in Fig. 3(a).

 

Fig. 3 Quantum-dot trimers with the strongest CD peaks for (a) γB + δE and [(c)–(f)] γA. Trimers in panels (c) through (e) correspond to three critical points of surface R1 A shown in panel (b). Transformation between the two optimal configurations, shown by the green curve in panel (b), is illustrated by the shaded regions in panel (f). The vertical scale in (b) is from −0.6 to 0.6 and the six contour lines are −0.5, −0.3, . . . , 0.5.

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Another kind of trimers that deserves special consideration is the one with ξ = ϑ. Such trimers have a doubly degenerate state of energy E1,2 = EA and a nondegenerate state of energy E3 = E + A. As we have seen earlier from Eq. (12), the intensity of the CD peaks produced by the population of these states is determined for γA by the product of the rotatory strength and the energy gap between the states, i.e. by R1 A. This product is plotted as a function of angles ϑ and φ in Fig. 3(b). Three kinds of optimal trimer configurations correspond to the following three types of critical points of this function [see Eqs. (41) and (44)]:

absolutemaximum:ϑ66.42°,φ=π;
localmaximum:ϑ040.20°,φ+53.73°;
saddlepoint:ϑ+21.33°,φ=0.
These configurations are shown in Figs. 3(c)–3(e). Much like in the case of the quantum-dot dimer, they result in almost equally strong CD peaks, with an approximate ratio of intensities 0.57:0.44:0.41 (see Table 2 in Appendix B). Since the weakest of these peaks — featured by the dimer in panel (e) — corresponds to the saddle point of surface R1 A, there is a continuum of optimal trimer configurations, the parameters of some of which are shown by the green curve in Fig. 3(b). These configurations are shaded in yellow in panel (f) and produce CD peaks with intensities between the intensities of the CD peaks of dimers (d) and (e).

The analysis of trimers with ξ ≠ ±ϑ is more complicated and can be performed using the general expressions given by Eqs. (33) through (38) in Appendix B.

5. Quantum-dot tetramer

As the last example, we study the optical activity of four coupled quantum dots located in the vertices of a tetrahedron shown in Fig. 1(c). The base of the tetrahedron is an equilateral triangle formed by the quantum dots whose dipole moments are assumed to be in the plane of this base. The azimuths of dipoles d1, d2, and d3 are α + 2π/3, α − 2π/3, and α respectively, and the direction of the fourth dipole is set by the polar angle ϑ and azimuth φ. The energies and wave functions of this tetramer states have the simplest forms for ϑ = 0 and ϑ = π/2. These two special cases are considered separately in Appendix C.

Tetramers with ϑ = 0 have two nondegenerate optically active states of energies E1,2 = EδE±, and two doubly degenerate optically inactive states of energies E3,4 = E + A, where δE±=A±A2+3B2 and coefficients A and B are given in Eq. (50). The rotatory strengths of the nondegenerate states are proportional to sin 2α, vanishing for achiral tetramers with α = ±π/2, as well for chiral ones with α = 0 and ±π — in which the fourth dipole decoupled from the rest three. Figure 4(a) shows how EjE and Rj vary with α. The rotatory strengths heavily depend on height b of the tetrahedron, turning zero when either b = 0 or b → ∞ due to the absence of coupling between the dipoles in the basic triangle and in the vertex. It is easy to show using Eqs. (50) and (54) that the strength of this interaction peaks for b=a/(23), where a is the distance between the quantum dots in the base, and that so do the rotatory strengths and the product R1(δE+δE), which determines the intensities of relatively broad lines in the CD spectrum. Since R1(δE+δE) ∝ sin 2α, the CD signal is also the strongest for tetramers with α = ±π/4 or ±3π/4.

 

Fig. 4 Energies (upper panels) of four tetramer states and rotatory strengths (lower panels) of the respective interband transitions as functions of azimuth α for a = b and different polar angles ϑ, which determine orientations of the quantum dots constituting the tetramer [see Fig. 1(c)]. The azimuth φ of the fourth dipole affects neither energies nor rotatory strengths.

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When the direction of the fourth dipole deflects from the vertical, the degeneracy of states Ψ3 and Ψ4 is lifted and the fourth state, too, becomes optically active. Figures 4(b) through 4(g) show how the α-spectra of energies and rotatory strengths transform with ϑ. Note that the shapes of these spectra are independent of the azimuth φ of the fourth dipole, which affects only the wave functions of the tetramer states. Four anticrossings in the energy spectra of states Ψ1, Ψ2, and Ψ4 are seen to result in the renormalization of the rotatory strengths’ spectra. As ϑ increases from zero to π/4, four anticrossings merge into two and then split back into four when ϑ is increased still further. At the same time, the rotatory strengths change drastically, and spectra R2(α) and R4(α) swap their places. It should also be noted that the variation of ϑ neither affects the energy of state Ψ3 nor makes this state optically active.

Figure 4(h) shows the energies and rotatory strengths of four nondegenerate states of the tetramer with ϑ = π/2. The states’ energies are E1 = E − 2A, E2 = E + δE, E3 = E + A, and E4 = E + δE+, where δE±=A/2±(A/2)2+2(B2+3C2) and coefficients A, B, and C are given in Eqs. (50), (56), and (57). The optically active states are Ψ2 and Ψ4. Much like in the case of ϑ = 0, the rotatory strengths vanish for α = 0, ±π/2, and ±π. There is also an optimal tetrahedron height, because R2b for b → 0 and R2 ∝ 1/b2 for b → ∞, but its depends on α and does not have a simple analytical expression. From Eqs. (56), (57), (59), and (64) it follows that R2(δE+δE) ∝ sin 2α, so that the strongest CD signal is achieved for tetramers with α = ±π/4 or ±3π/4. The same equations also show that the CD signal is the strongest for b=a/(23).

6. Conclusion

We have developed an elegant theoretical model which allows one to calculate the rotatory strengths and dissymmetry factors of optical transitions upon the excitation of the lowest-order states of quantum-dot molecules. This model was used to analytically study the optical activity of molecules composed of two, three, and four semiconductor nanocrystals. We found the optimal geometries of such dimers, trimers, and tetramers resulting in the strongest CD signals when the widths of the CD peaks significantly exceed the energy gaps between them. It was also shown that the rotatory strengths of quantum-dot molecules can exceed the typical rotatory strengths of naturally occurring chiral molecules by five to six orders of magnitude. The derived analytical expressions and the discovered optimal geometries are useful for engineering optically active quantum-dot materials and devices for advanced nanophotonics applications.

Appendix A: Quantum-dot dimer

The positions and dipole moments of the two quantum dots in Fig. 1(a) are given by r1 = (0, 0, 0), r2 = (0, 0, a), and dj = d (sin ϑj cos φj, sin ϑj sin φj, cos ϑj). The energies of the two molecular states and the coefficients in Eq. (3) are found to be give by

E1=EδE,E2=E+δE,c1=12(1,1),c2=12(1,1),
δE=(sinϑ1sinϑ2cosδφ2cosϑ1cosϑ2)η,δφ=φ1φ2,η=d2/(εa3).
The rotatory strengths and dissymmetry factors are given by
R1=R2=ρsinϑ1sinϑ2sinδφ,
g1,2=4χsinϑ1sinϑ2sinδφ2(sinϑ1sinϑ2cosδφ+cosϑ1cosϑ21)χ2sin2ϑ2,
where ρ = (χ/2) d2, χ = , and the minus or plus sign corresponds to state 1 or 2.

Optimal ϑ1, ϑ2, and δφ for broad CD peaks

In the case of broad spectral lines, the conditions of the extrema of the CD signal — (R1δE)ϑ1=0, (R1δE)ϑ2=0, and (R1δE)δφ=0 — lead to the following system of equations:

2cos2ϑ1cotϑ2=cosδφsin2ϑ1,
2cos2ϑ2cotϑ1=cosδφsin2ϑ2,
2cosδφcotϑ1cotϑ2=cos2δφ,
in deriving which we have taken into account that R1 ≠ 0.

To solve this system under the condition R1δE ≠ 0, we notice that the last equation requires that cos δφ ≠ 0. For cos 2δφ = 0 this equation reduces to cos ϑ1 cos ϑ2 = 0 and, together with the first pair of equations, gives cos ϑ1 = cos ϑ2 = 0. Taking into account that δφ is between −π and π, we thus get the first set of critical points

δφ=±π4,ϑ1=ϑ2=π2.

Because cos δφ ≠ 0, Eqs. (22) and (23) require that cos 2ϑ1 ≠ 0 and cos 2ϑ2 ≠ 0. Using the interchangeability of ϑ1 and ϑ2 in our system and assuming that all the multipliers in it are nonzero, we can eliminate cot ϑ2 from Eqs. (22) and (24) to get cos 2ϑ1 = cos 2ϑ2 = 1 − sin−2 δφ. This result reduces Eq. (24) to (2cosδφ+1)(2cosδφ+1)=0, leading to another five sets of critical points

δφ=±2π3,ϑ1,2={12arccos(13),π12arccos(13)};
δφ=±3π4,ϑ1=ϑ2=π2.

The obtained six sets of critical points are given in Table 1. One can see that two absolute maxima and two absolute minima R1δE=±w/3, where w = ρη, are achieved for δφ = ±2π/3 and ϑ1 = ϑ2 = (1/2) arccos(−1/3). There is also another pair of optimal dimer configurations, with δφ = {±π/4, ±3π/4} and ϑ1 = ϑ2 = π/2, which have close values of peak CD signals corresponding to R1δE = ±w/2. Note that the dimers with different signs of δφ are mirror images of each other, and that ϑ1 = ϑ2 = ϑ0 and ϑ1 = ϑ2 = πϑ0 describe the same dimer.

Tables Icon

Table 1. Critical points of function R1δE; ϑ0 = (1/2) arccos(−1/3).

Appendix B: Quantum-dot trimer

The trimer shown in Fig. 1(b) is described by the parameters

r1=(0,a32,0),r2=(a2,0,0),r3=(a2,0,0),
d1=d(sinξcosφ,sinξsinφ,cosξ),
d2=d(sinϑcos(φ+2π/3),sinϑsin(φ+2π/3),cosϑ),
d3=d(sinϑcos(φ2π/3),sinϑsin(φ2π/3),cosϑ).
The Hamiltonian of this molecule is of the form
(EAAAEBABE),
where
A=(cosϑcosξ14(6cos2φ1)sinϑsinξ)η,B=(134(1+2cos2φ)sin2ϑ)η.
The solution to the eigensystem problem is given by
E1=EB,E2=E+δE,E3=E+δE+,δE±=12(B±8A2+B2),
c1=12(0,1,1),c2=(δE+,A,A)(δE+2+2A2)1/2,c3=(δE,A,A)(δE2+2A2)1/2,
and the rotatory strengths are
R1=ρ32sin2ϑcosφ,
R2=ρ3A2δE+sin(ϑ+ξ)Asin2ϑ4A2+BδE+cosφ,
R3=ρ3A2δEsin(ϑ+ξ)Asin2ϑ4A2+BδEcosφ.
It is easy to verify that R1 + R2 + R3 = 0 and that for ϑ = ξ we have R1 = R2 = −R3/2.

Optimal ϑ and φ for broad CD peaks

For ϑ = ξ we have A = B, and the strongest CD signal is achieved when product R1 A is maximal. In this case the critical trimer parameters ϑ and φ obey the system of equations

(3cos3φ2cosφ)cos2ϑ=3(cos3φ+2cosφ)cos4ϑ,
3(3sin3φ+2sinφ)sin2ϑ=4sinφ.
First of all, we notice that a simple solution to this system — corresponding to sin φ = 0 and 9 cos 4ϑ = cos 2ϑ — is given by
φ={0,π},ϑ={ϑ±,πϑ±},ϑ±=12arccos(1±64936).
Next, for sin φ ≠ 0 this system reduces to
20cos2φ=7,
12sin2ϑ=5,
giving the following critical angles:
φ={φ±,2πφ±},ϑ={ϑ0,πϑ0},
φ±=arccos(±1275),ϑ0=arcsin(1253).

Equations (41), (44), and (45) give 16 critical points of function R1 A in the domain 0 ≤ φ < 2π and 0 ≤ ϑπ [see Fig. 3(b)]. Table 2 shows six different types of these critical points and the associated values of rotatory strength, energy parameter, and their product. The rest of critical points belong to one of these types, representing either dimers whose parameters are given in the table or their mirror-image enantiomers.

Tables Icon

Table 2. Critical points of function R1 A; angles ϑ±, φ±, and ϑ0 are defined in Eqs. (41) and (45).

Appendix C: Quantum-dot tetramer

The positions and dipole moments of the quantum dots comprising the tetramer shown in Fig. 1(c) are given by

r1=(a2,a23,0),r2=(a2,a23,0),r3=(0,a3,0),r4=(0,0,b),
d1=d(cos(α+2π/3),sin(α+2π/3),0),d2=d(cos(α2π/3),sin(α2π/3),0),
d3=d(cosα,sinα,0),d4=d(sinϑcosφ,sinϑsinφ,cosϑ).
There are two special cases in which the rotatory strengths of optical transitions are given by simple formulas.

The special case of ϑ = 0

For ϑ = 0 the Hamiltonian of the tetramer is of the form

(EAABAEABAAEBBBBE),
where
A=14(6cos2α1)η,B=27a4bsinα(a2+3b2)5/2η.
The solution to the eigensystem problem is given by
E1=EδE+,E2=EδE,E3=E4=E+A,δE±=A±A2+3B2,
c1=(δE+,δE+,δE+,3B)(3δE+2+9B2)1/2,c2=(δE,δE,δE,3B)(3δE2+9B2)1/2,
c3=12(1,0,1,0),c4=12(1,1,0,0),
and the rotatory strengths are
R1=R2=ρ3BA2+3B2cosα,R3=R4=0.

The special case of ϑ = π/2

For ϑ = π/2 the Hamiltonian of the tetramer is given by the matrix

(EAAB+CAEAB+CAAE2CB+CB+C2CE),
where A is given in Eq. (50),
B=9a3(a26b2)sin(αφ)+3a2sin(α+φ)4(a2+3b2)5/2η,
C=33a3(a26b2)cos(αφ)3a2cos(α+φ)4(a2+3b2)5/2η.
The solution to the eigensystem problem is given by
E1=E2A,E2=E+δE,E3=E+A,E4=E+δE+,
δE±=12(A±A2+8(B2+3C2)),
c1=13(1,1,1,0),
c2=1N+(B+C,B+C,2C,δE+),
c3=16(B2+3C2)(B3C,B+3C,2B,0),
c4=1N(B+C,B+C,2C,δE),
where N±=2B3+6C2+δE±2. The rotatory strengths are
R1=R3=0,R2=R4=σ3Bcos(αφ)3Csin(αφ)A2+8(B2+3C2)bd2.
It can be shown that B2 + 3C2 and 3Bcos(αφ)3Csin(αφ) are independent of φ, so that the energy spectrum and rotatory strengths can be calculated by setting φ = 0.

Funding

Ministry of Education and Science of the Russian Federation (14.B25.31.0002).

Acknowledgments

A.S.B. and I.D.R. gratefully acknowledge the financial support from the Ministry of Education and Science of the Russian Federation through its Scholarship of the President of the Russian Federation for young scientists and its Grant of the President of the Russian Federation for young scientists.

References and links

1. J. Wu and Z. M. Wang, Quantum Dot Molecules (Springer, 2014). [CrossRef]  

2. A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013). [CrossRef]  

3. K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012). [CrossRef]  

4. S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012). [CrossRef]  

5. K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012). [CrossRef]   [PubMed]  

6. E. Rahimi and S. M. Nejad, “Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates,” Nanoscale Res. Lett. 7, 274 (2012). [CrossRef]   [PubMed]  

7. D. De, K. P. Ghatak, and S. Bhattacharaya, Quantum Dots and Quantum Cellular Automata (Nova Science Publishers, Inc, 2013).

8. Y. He, W. Bo, R. K. Dukor, and L. A. Nafie, “Determination of absolute configuration of chiral molecules using vibrational optical activity: A review,” Appl. Spectrosc. 65, 699–723 (2011). [CrossRef]   [PubMed]  

9. M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008). [CrossRef]   [PubMed]  

10. N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016). [CrossRef]  

11. A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015). [CrossRef]   [PubMed]  

12. M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015). [CrossRef]  

13. J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005). [CrossRef]   [PubMed]  

14. A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016). [CrossRef]  

15. S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015). [CrossRef]  

16. M. V. Mukhina, V. G. Maslov, A. V. Baranov, M. V. Artemyev, A. O. Orlova, and A. V. Fedorov, “Anisotropy of optical transitions in ordered ensemble of CdSe quantum rods,” Opt. Lett. 38, 3426–3428 (2013). [CrossRef]   [PubMed]  

17. A. S. Baimuratov, I. D. Rukhlenko, and A. V. Fedorov, “Engineering band structure in nanoscale quantum-dot supercrystals,” Opt. Lett. 38, 2259–2261 (2013). [CrossRef]   [PubMed]  

18. A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015). [CrossRef]   [PubMed]  

19. N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016). [CrossRef]  

20. I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016). [CrossRef]   [PubMed]  

21. X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015). [CrossRef]   [PubMed]  

22. G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014). [CrossRef]   [PubMed]  

23. N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012). [CrossRef]   [PubMed]  

24. A. O. Govorov and Z. Fan, “Theory of chiral plasmonic nanostructures comprising metal nanocrystals and chiral molecular media,” ChemPhysChem 13, 2551–2560 (2012). [CrossRef]   [PubMed]  

25. A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012). [CrossRef]   [PubMed]  

26. B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011). [CrossRef]   [PubMed]  

27. A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010). [CrossRef]   [PubMed]  

28. Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10, 2580–2587 (2010). [CrossRef]   [PubMed]  

29. A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016). [CrossRef]  

30. A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014). [CrossRef]   [PubMed]  

31. N. Berova, Comprehensive Chiroptical Spectroscopy, vol. 1 (John Wiley & Sons, 2012). [CrossRef]  

32. L. Rosenfeld, “Quantenmechanische theorie der natürlichen optischen aktivität von flüssigkeiten und gasen,” Z. Phys. 52, 161–174 (1929). [CrossRef]  

33. M. Nieto-Vesperinas, “Resonant chirality in light scattered from magnetodielectric particles,” arXiv preprint arXiv:1611.02613 (2016).

34. M. Nieto-Vesperinas, “Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism,” Phys. Rev. A 92, 023813 (2015). [CrossRef]  

35. X. Zambrana-Puyalto and N. Bonod, “Tailoring the chirality of light emission with spherical Si-based antennas,” Nanoscale 8, 10441–10452 (2016). [CrossRef]   [PubMed]  

36. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010). [CrossRef]   [PubMed]  

37. Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009). [CrossRef]   [PubMed]  

38. R. R. Judkins and D. J. Royer, “Optical rotatory strength of tris-bidentate cobalt(III) complexes,” Inorg. Chem. 13, 945–950 (1974). [CrossRef]  

39. T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971). [CrossRef]   [PubMed]  

40. P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969). [CrossRef]   [PubMed]  

41. I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express 19, 15459–15482 (2011). [CrossRef]   [PubMed]  

42. M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006). [CrossRef]   [PubMed]  

References

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  1. J. Wu and Z. M. Wang, Quantum Dot Molecules (Springer, 2014).
    [Crossref]
  2. A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
    [Crossref]
  3. K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
    [Crossref]
  4. S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
    [Crossref]
  5. K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
    [Crossref] [PubMed]
  6. E. Rahimi and S. M. Nejad, “Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates,” Nanoscale Res. Lett. 7, 274 (2012).
    [Crossref] [PubMed]
  7. D. De, K. P. Ghatak, and S. Bhattacharaya, Quantum Dots and Quantum Cellular Automata (Nova Science Publishers, Inc, 2013).
  8. Y. He, W. Bo, R. K. Dukor, and L. A. Nafie, “Determination of absolute configuration of chiral molecules using vibrational optical activity: A review,” Appl. Spectrosc. 65, 699–723 (2011).
    [Crossref] [PubMed]
  9. M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008).
    [Crossref] [PubMed]
  10. N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
    [Crossref]
  11. A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
    [Crossref] [PubMed]
  12. M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
    [Crossref]
  13. J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
    [Crossref] [PubMed]
  14. A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
    [Crossref]
  15. S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
    [Crossref]
  16. M. V. Mukhina, V. G. Maslov, A. V. Baranov, M. V. Artemyev, A. O. Orlova, and A. V. Fedorov, “Anisotropy of optical transitions in ordered ensemble of CdSe quantum rods,” Opt. Lett. 38, 3426–3428 (2013).
    [Crossref] [PubMed]
  17. A. S. Baimuratov, I. D. Rukhlenko, and A. V. Fedorov, “Engineering band structure in nanoscale quantum-dot supercrystals,” Opt. Lett. 38, 2259–2261 (2013).
    [Crossref] [PubMed]
  18. A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
    [Crossref] [PubMed]
  19. N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
    [Crossref]
  20. I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
    [Crossref] [PubMed]
  21. X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015).
    [Crossref] [PubMed]
  22. G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
    [Crossref] [PubMed]
  23. N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
    [Crossref] [PubMed]
  24. A. O. Govorov and Z. Fan, “Theory of chiral plasmonic nanostructures comprising metal nanocrystals and chiral molecular media,” ChemPhysChem 13, 2551–2560 (2012).
    [Crossref] [PubMed]
  25. A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
    [Crossref] [PubMed]
  26. B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011).
    [Crossref] [PubMed]
  27. A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
    [Crossref] [PubMed]
  28. Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10, 2580–2587 (2010).
    [Crossref] [PubMed]
  29. A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
    [Crossref]
  30. A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
    [Crossref] [PubMed]
  31. N. Berova, Comprehensive Chiroptical Spectroscopy, vol. 1 (John Wiley & Sons, 2012).
    [Crossref]
  32. L. Rosenfeld, “Quantenmechanische theorie der natürlichen optischen aktivität von flüssigkeiten und gasen,” Z. Phys. 52, 161–174 (1929).
    [Crossref]
  33. M. Nieto-Vesperinas, “Resonant chirality in light scattered from magnetodielectric particles,” arXiv preprint arXiv:1611.02613 (2016).
  34. M. Nieto-Vesperinas, “Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism,” Phys. Rev. A 92, 023813 (2015).
    [Crossref]
  35. X. Zambrana-Puyalto and N. Bonod, “Tailoring the chirality of light emission with spherical Si-based antennas,” Nanoscale 8, 10441–10452 (2016).
    [Crossref] [PubMed]
  36. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
    [Crossref] [PubMed]
  37. Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009).
    [Crossref] [PubMed]
  38. R. R. Judkins and D. J. Royer, “Optical rotatory strength of tris-bidentate cobalt(III) complexes,” Inorg. Chem. 13, 945–950 (1974).
    [Crossref]
  39. T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971).
    [Crossref] [PubMed]
  40. P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969).
    [Crossref] [PubMed]
  41. I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express 19, 15459–15482 (2011).
    [Crossref] [PubMed]
  42. M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
    [Crossref] [PubMed]

2016 (6)

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref] [PubMed]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

X. Zambrana-Puyalto and N. Bonod, “Tailoring the chirality of light emission with spherical Si-based antennas,” Nanoscale 8, 10441–10452 (2016).
[Crossref] [PubMed]

2015 (6)

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

M. Nieto-Vesperinas, “Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism,” Phys. Rev. A 92, 023813 (2015).
[Crossref]

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

2014 (2)

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

2013 (3)

2012 (7)

K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
[Crossref]

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

E. Rahimi and S. M. Nejad, “Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates,” Nanoscale Res. Lett. 7, 274 (2012).
[Crossref] [PubMed]

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

A. O. Govorov and Z. Fan, “Theory of chiral plasmonic nanostructures comprising metal nanocrystals and chiral molecular media,” ChemPhysChem 13, 2551–2560 (2012).
[Crossref] [PubMed]

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (3)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10, 2580–2587 (2010).
[Crossref] [PubMed]

2009 (1)

Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009).
[Crossref] [PubMed]

2008 (1)

M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008).
[Crossref] [PubMed]

2006 (1)

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
[Crossref] [PubMed]

2005 (1)

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
[Crossref] [PubMed]

1974 (1)

R. R. Judkins and D. J. Royer, “Optical rotatory strength of tris-bidentate cobalt(III) complexes,” Inorg. Chem. 13, 945–950 (1974).
[Crossref]

1971 (1)

T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971).
[Crossref] [PubMed]

1969 (1)

P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969).
[Crossref] [PubMed]

1929 (1)

L. Rosenfeld, “Quantenmechanische theorie der natürlichen optischen aktivität von flüssigkeiten und gasen,” Z. Phys. 52, 161–174 (1929).
[Crossref]

Abdulrahman, N. A.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

Albelda, M. T.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
[Crossref] [PubMed]

Alonso-Gomez, J. L.

B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011).
[Crossref] [PubMed]

Arakawa, Y.

K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
[Crossref]

Artemyev, M. V.

Auguie, B.

B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011).
[Crossref] [PubMed]

Badolato, A.

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Baimuratov, A. S.

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, and A. V. Fedorov, “Engineering band structure in nanoscale quantum-dot supercrystals,” Opt. Lett. 38, 2259–2261 (2013).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

Baranov, A. V.

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref] [PubMed]

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, M. V. Artemyev, A. O. Orlova, and A. V. Fedorov, “Anisotropy of optical transitions in ordered ensemble of CdSe quantum rods,” Opt. Lett. 38, 3426–3428 (2013).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

Baskin, A.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Bayley, P. M.

T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971).
[Crossref] [PubMed]

P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969).
[Crossref] [PubMed]

Baymuratov, A. S.

Berova, N.

N. Berova, Comprehensive Chiroptical Spectroscopy, vol. 1 (John Wiley & Sons, 2012).
[Crossref]

Berwick, K.

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

Bhattacharaya, S.

D. De, K. P. Ghatak, and S. Bhattacharaya, Quantum Dots and Quantum Cellular Automata (Nova Science Publishers, Inc, 2013).

Bo, W.

Bonod, N.

X. Zambrana-Puyalto and N. Bonod, “Tailoring the chirality of light emission with spherical Si-based antennas,” Nanoscale 8, 10441–10452 (2016).
[Crossref] [PubMed]

Bordel, D.

K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
[Crossref]

Bracker, A. S.

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Canary, J. W.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
[Crossref] [PubMed]

Chan, H.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Climente, J. I.

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Cohen, A. E.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009).
[Crossref] [PubMed]

Cook, T. A.

Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009).
[Crossref] [PubMed]

De, D.

D. De, K. P. Ghatak, and S. Bhattacharaya, Quantum Dots and Quantum Cellular Automata (Nova Science Publishers, Inc, 2013).

Delley, Y. L.

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

Doty, M. F.

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Dukor, R. K.

Economou, S. E.

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Elzerman, J. M.

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

Fan, Z.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

A. O. Govorov and Z. Fan, “Theory of chiral plasmonic nanostructures comprising metal nanocrystals and chiral molecular media,” ChemPhysChem 13, 2551–2560 (2012).
[Crossref] [PubMed]

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10, 2580–2587 (2010).
[Crossref] [PubMed]

Fang, Y.

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015).
[Crossref] [PubMed]

Fedorov, A. V.

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref] [PubMed]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

M. V. Mukhina, V. G. Maslov, A. V. Baranov, M. V. Artemyev, A. O. Orlova, and A. V. Fedorov, “Anisotropy of optical transitions in ordered ensemble of CdSe quantum rods,” Opt. Lett. 38, 3426–3428 (2013).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, and A. V. Fedorov, “Engineering band structure in nanoscale quantum-dot supercrystals,” Opt. Lett. 38, 2259–2261 (2013).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express 19, 15459–15482 (2011).
[Crossref] [PubMed]

Gadegaard, N.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

Gammon, D.

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Gelman, E.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Ghatak, K. P.

D. De, K. P. Ghatak, and S. Bhattacharaya, Quantum Dots and Quantum Cellular Automata (Nova Science Publishers, Inc, 2013).

Ginzburg, P.

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

Govan, J.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

Govorov, A. O.

A. O. Govorov and Z. Fan, “Theory of chiral plasmonic nanostructures comprising metal nanocrystals and chiral molecular media,” ChemPhysChem 13, 2551–2560 (2012).
[Crossref] [PubMed]

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10, 2580–2587 (2010).
[Crossref] [PubMed]

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Graham, M. W.

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
[Crossref] [PubMed]

Guerrero-Martinez, A.

B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011).
[Crossref] [PubMed]

Guimard, D.

K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
[Crossref]

Gun’ko, Y. K.

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

He, Y.

Hendry, E.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

Hernandez, P.

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Högele, A.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Imamoglu, A.

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

Judkins, R. R.

R. R. Judkins and D. J. Royer, “Optical rotatory strength of tris-bidentate cobalt(III) complexes,” Inorg. Chem. 13, 945–950 (1974).
[Crossref]

Kadodwala, M.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

Kelly, S. M.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

King, T. S.

T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971).
[Crossref] [PubMed]

Klajn, R.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Kral, P.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Kruchinin, S. Y.

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

Kuzyk, A.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Leonov, M. Y.

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

Liedl, T.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Liu, Y.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
[Crossref] [PubMed]

Liz-Marzan, L. M.

B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011).
[Crossref] [PubMed]

Loudon, A.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

Maslov, V. G.

Miguel-Sanchez, J.

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

Moloney, M. P.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

Mukhina, M.

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

Mukhina, M. V.

Nafie, L. A.

Naik, R. R.

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Nejad, S. M.

E. Rahimi and S. M. Nejad, “Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates,” Nanoscale Res. Lett. 7, 274 (2012).
[Crossref] [PubMed]

Nielsen, E. B.

P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, “Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism,” Phys. Rev. A 92, 023813 (2015).
[Crossref]

M. Nieto-Vesperinas, “Resonant chirality in light scattered from magnetodielectric particles,” arXiv preprint arXiv:1611.02613 (2016).

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, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

Orlova, A. O.

Pardatscher, G.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Perova, T. S.

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

Ponomareva, I. O.

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

Premaratne, M.

Quack, M.

M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008).
[Crossref] [PubMed]

Rahimi, E.

E. Rahimi and S. M. Nejad, “Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates,” Nanoscale Res. Lett. 7, 274 (2012).
[Crossref] [PubMed]

Repnin, N.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Roller, E.-M.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Rosenfeld, L.

L. Rosenfeld, “Quantenmechanische theorie der natürlichen optischen aktivität von flüssigkeiten und gasen,” Z. Phys. 52, 161–174 (1929).
[Crossref]

Royer, D. J.

R. R. Judkins and D. J. Royer, “Optical rotatory strength of tris-bidentate cobalt(III) complexes,” Inorg. Chem. 13, 945–950 (1974).
[Crossref]

Rukhlenko, I. D.

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref] [PubMed]

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, and A. V. Fedorov, “Engineering band structure in nanoscale quantum-dot supercrystals,” Opt. Lett. 38, 2259–2261 (2013).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

I. D. Rukhlenko, A. V. Fedorov, A. S. Baymuratov, and M. Premaratne, “Theory of quasi-elastic secondary emission from a quantum dot in the regime of vibrational resonance,” Opt. Express 19, 15459–15482 (2011).
[Crossref] [PubMed]

Salvador, M. R.

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
[Crossref] [PubMed]

Scholes, G. D.

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
[Crossref] [PubMed]

Schreiber, R.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Shellman, J. A.

P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969).
[Crossref] [PubMed]

Simmel, F. C.

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Singh, G.

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Slocik, J. M.

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Stohner, J.

M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008).
[Crossref] [PubMed]

Sun, M.

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015).
[Crossref] [PubMed]

Tanabe, K.

K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
[Crossref]

Tang, Y.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009).
[Crossref] [PubMed]

Tepliakov, N. V.

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

I. D. Rukhlenko, A. S. Baimuratov, N. V. Tepliakov, A. V. Baranov, and A. V. Fedorov, “Shape-induced optical activity of chiral nanocrystals,” Opt. Lett. 41, 2438–2441 (2016).
[Crossref] [PubMed]

Tian, X.

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015).
[Crossref] [PubMed]

Tonooka, T.

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

Turkov, V. K.

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

Wang, Z. M.

J. Wu and Z. M. Wang, Quantum Dot Molecules (Springer, 2014).
[Crossref]

Weiss, K. M.

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

Willeke, M.

M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008).
[Crossref] [PubMed]

Wu, J.

J. Wu and Z. M. Wang, Quantum Dot Molecules (Springer, 2014).
[Crossref]

Yong, F. C.

T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971).
[Crossref] [PubMed]

Zambrana-Puyalto, X.

X. Zambrana-Puyalto and N. Bonod, “Tailoring the chirality of light emission with spherical Si-based antennas,” Nanoscale 8, 10441–10452 (2016).
[Crossref] [PubMed]

Zhang, J.

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
[Crossref] [PubMed]

Annu. Rev. Phys. Chem. (1)

M. Quack, J. Stohner, and M. Willeke, “High-resolution spectroscopic studies and theory of parity violation in chiral molecules,” Annu. Rev. Phys. Chem. 59, 741–769 (2008).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

K. Tanabe, D. Guimard, D. Bordel, and Y. Arakawa, “High-efficiency InAs/GaAs quantum dot solar cells by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 100, 193905 (2012).
[Crossref]

Appl. Spectrosc. (1)

ChemPhysChem (1)

A. O. Govorov and Z. Fan, “Theory of chiral plasmonic nanostructures comprising metal nanocrystals and chiral molecular media,” ChemPhysChem 13, 2551–2560 (2012).
[Crossref] [PubMed]

Chirality (1)

J. Zhang, M. T. Albelda, Y. Liu, and J. W. Canary, “Chiral nanotechnology,” Chirality 17, 404–420 (2005).
[Crossref] [PubMed]

Eur. J. Biochem. (1)

T. S. King, P. M. Bayley, and F. C. Yong, “Optical rotatory dispersion and circular dichroism of cytochrome oxidase,” Eur. J. Biochem. 20, 103–110 (1971).
[Crossref] [PubMed]

Inorg. Chem. (1)

R. R. Judkins and D. J. Royer, “Optical rotatory strength of tris-bidentate cobalt(III) complexes,” Inorg. Chem. 13, 945–950 (1974).
[Crossref]

J. Appl. Phys. (2)

S. Y. Kruchinin, I. D. Rukhlenko, A. S. Baimuratov, M. Y. Leonov, V. K. Turkov, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Photoluminescence of a quantum-dot molecule,” J. Appl. Phys. 117, 014306 (2015).
[Crossref]

N. V. Tepliakov, A. S. Baimuratov, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Optical activity of chirally distorted nanocrystals,” J. Appl. Phys. 119, 194302 (2016).
[Crossref]

J. Chem. Phys. (1)

M. R. Salvador, M. W. Graham, and G. D. Scholes, “Exciton-phonon coupling and disorder in the excited states of CdSe colloidal quantum dots,” J. Chem. Phys. 125, 184709 (2006).
[Crossref] [PubMed]

J. Phys. Chem. (1)

P. M. Bayley, E. B. Nielsen, and J. A. Shellman, “The rotatory properties of molecules containing two peptide groups: Theory,” J. Phys. Chem. 73, 228–243 (1969).
[Crossref] [PubMed]

J. Phys. Chem. A (1)

Y. Tang, T. A. Cook, and A. E. Cohen, “Limits on fluorescence detected circular dichroism of single helicene molecules,” J. Phys. Chem. A 113, 6213–6216 (2009).
[Crossref] [PubMed]

J. Phys. Chem. Lett. (1)

B. Auguie, J. L. Alonso-Gomez, A. Guerrero-Martinez, and L. M. Liz-Marzan, “Fingers crossed: Optical activity of a chiral dimer of plasmonic nanorods,” J. Phys. Chem. Lett. 2, 846–851 (2011).
[Crossref] [PubMed]

Nano Lett. (4)

A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: Plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10, 1374–1382 (2010).
[Crossref] [PubMed]

Z. Fan and A. O. Govorov, “Plasmonic circular dichroism of chiral metal nanoparticle assemblies,” Nano Lett. 10, 2580–2587 (2010).
[Crossref] [PubMed]

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12, 977–983 (2012).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Dislocation-induced chirality of semiconductor nanocrystals,” Nano Lett. 15, 1710–1715 (2015).
[Crossref] [PubMed]

Nanophotonics (1)

N. V. Tepliakov, A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Engineering optical activity of semiconductor nanocrystals via ion doping,” Nanophotonics 5, 517–522 (2016).
[Crossref]

Nanoscale (1)

X. Zambrana-Puyalto and N. Bonod, “Tailoring the chirality of light emission with spherical Si-based antennas,” Nanoscale 8, 10441–10452 (2016).
[Crossref] [PubMed]

Nanoscale Res. Lett. (1)

E. Rahimi and S. M. Nejad, “Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates,” Nanoscale Res. Lett. 7, 274 (2012).
[Crossref] [PubMed]

Nature (1)

A. Kuzyk, R. Schreiber, Z. Fan, G. Pardatscher, E.-M. Roller, A. Högele, F. C. Simmel, A. O. Govorov, and T. Liedl, “DNA-based self-assembly of chiral plasmonic nanostructures with tailored optical response,” Nature 483, 311–314 (2012).
[Crossref] [PubMed]

Nature Protoc. (1)

M. P. Moloney, J. Govan, A. Loudon, M. Mukhina, and Y. K. Gun’ko, “Preparation of chiral quantum dots,” Nature Protoc. 10, 558–573 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

M. Nieto-Vesperinas, “Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism,” Phys. Rev. A 92, 023813 (2015).
[Crossref]

Phys. Rev. B (1)

S. E. Economou, J. I. Climente, A. Badolato, A. S. Bracker, D. Gammon, and M. F. Doty, “Scalable qubit architecture based on holes in quantum dot molecules,” Phys. Rev. B 86, 085319 (2012).
[Crossref]

Phys. Rev. Lett. (2)

K. M. Weiss, J. M. Elzerman, Y. L. Delley, J. Miguel-Sanchez, and A. Imamoğlu, “Coherent two-electron spin qubits in an optically active pair of coupled InGaAs quantum dots,” Phys. Rev. Lett. 109, 107401 (2012).
[Crossref] [PubMed]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Sci. Rep. (6)

A. S. Baimuratov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Chiral quantum supercrystals with total dissymetry of optical response,” Sci. Rep. 6, 23321 (2016).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, I. O. Ponomareva, M. Y. Leonov, T. S. Perova, K. Berwick, A. V. Baranov, and A. V. Fedorov, “Level anticrossing of impurity states in semiconductor nanocrystals,” Sci. Rep. 4, 6917 (2014).
[Crossref] [PubMed]

A. S. Baimuratov, I. D. Rukhlenko, V. K. Turkov, A. V. Baranov, and A. V. Fedorov, “Quantum-dot supercrystals for future nanophotonics,” Sci. Rep. 3, 1727 (2013).
[Crossref]

A. S. Baimuratov, I. D. Rukhlenko, R. E. Noskov, P. Ginzburg, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Giant optical activity of quantum dots, rods, and disks with screw dislocations,” Sci. Rep. 5, 14712 (2015).
[Crossref] [PubMed]

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5, 17534 (2015).
[Crossref] [PubMed]

A. S. Baimuratov, N. V. Tepliakov, Y. K. Gun’ko, A. V. Baranov, A. V. Fedorov, and I. D. Rukhlenko, “Mixing of quantum states: A new route to creating optical activity,” Sci. Rep. 6, 17 (2016).
[Crossref]

Science (1)

G. Singh, H. Chan, A. Baskin, E. Gelman, N. Repnin, P. Kral, and R. Klajn, “Self-assembly of magnetite nanocubes into helical superstructures,” Science 345, 1149–1153 (2014).
[Crossref] [PubMed]

Z. Phys. (1)

L. Rosenfeld, “Quantenmechanische theorie der natürlichen optischen aktivität von flüssigkeiten und gasen,” Z. Phys. 52, 161–174 (1929).
[Crossref]

Other (4)

M. Nieto-Vesperinas, “Resonant chirality in light scattered from magnetodielectric particles,” arXiv preprint arXiv:1611.02613 (2016).

N. Berova, Comprehensive Chiroptical Spectroscopy, vol. 1 (John Wiley & Sons, 2012).
[Crossref]

D. De, K. P. Ghatak, and S. Bhattacharaya, Quantum Dots and Quantum Cellular Automata (Nova Science Publishers, Inc, 2013).

J. Wu and Z. M. Wang, Quantum Dot Molecules (Springer, 2014).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Quantum-dot (a) dimer, (b) trimer, and (c) tetramer composed of equal but differently oriented semiconductor quantum dots. The molecules become optically active for certain orientations of dipole moments dj of the fundamental interband transitons inside the quantum dots; a is the distance between the quantum dots in the dimer, trimer, and in the base of the tetramer; b is the height of the tetramer.
Fig. 2
Fig. 2 Optimal configurations of quantum-dot dimers exhibiting the strongest peaks in the CD spectrum for γδE. The dipoles of the upper quantum dots are all in the yz plane. The intensities of peaks in the CD spectrum of dimer (a) exceed the intensities of peaks in the spectra of dimers (b) and (c) by a factor of 3 / 2. Panel (d) shows the transformation between the optimal configurations (c) and (a), which leads to a continuum of intermediate optimal orientations of the quantum-dot dipoles.
Fig. 3
Fig. 3 Quantum-dot trimers with the strongest CD peaks for (a) γB + δE and [(c)–(f)] γA. Trimers in panels (c) through (e) correspond to three critical points of surface R1 A shown in panel (b). Transformation between the two optimal configurations, shown by the green curve in panel (b), is illustrated by the shaded regions in panel (f). The vertical scale in (b) is from −0.6 to 0.6 and the six contour lines are −0.5, −0.3, . . . , 0.5.
Fig. 4
Fig. 4 Energies (upper panels) of four tetramer states and rotatory strengths (lower panels) of the respective interband transitions as functions of azimuth α for a = b and different polar angles ϑ, which determine orientations of the quantum dots constituting the tetramer [see Fig. 1(c)]. The azimuth φ of the fourth dipole affects neither energies nor rotatory strengths.

Tables (2)

Tables Icon

Table 1 Critical points of function R1δE; ϑ0 = (1/2) arccos(−1/3).

Tables Icon

Table 2 Critical points of function R1 A; angles ϑ±, φ±, and ϑ0 are defined in Eqs. (41) and (45).

Equations (64)

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V n ψ n ( r ) ψ m * ( r ) d r = δ nm ,
H = ( E V 12 V 1 N V 21 E V 2 N V N 1 V N 2 E ) .
Ψ μ = c μ 1 ψ 1 + c μ 2 ψ 2 + + c μ N ψ N c μ n ψ n .
R μ = e 2 ε 2 c E g m 2 Re ( p 0 μ [ r × p ] μ 0 ) ,
p μ 0 = Ψ μ | p ^ | 0 = c μ n * ψ n | p ^ | 0 = c μ n * p n
[ r × p ] μ 0 = Ψ μ | r ^ × p ^ | 0 = Ψ μ | r ^ × p n | ψ n = c μ m * ψ m | r ^ × p n | ψ n = c μ n * [ r n × p n ] ,
R μ = σ Re ( c μ n c μ m * ) ( d n [ r m × d m ] ) ,
D μ = | c μ n d n | 2 + σ 2 | c μ n [ r n × d n ] | 2 .
V nm = 1 ε r nm 5 [ r nm 2 ( d n d m ) 3 ( r nm d n ) ( r nm d n ) ] ,
δ E sin ϑ 1 sin ϑ 2 cos ( φ 2 φ 1 ) 2 cos ϑ 1 cos ϑ 2 ,
R 1 , 2 sin ϑ 1 sin ϑ 2 sin ( φ 1 φ 2 ) .
CD ( R 1 , δ E , γ , ω ) R 1 Γ ( E 1 , ω ) + R 2 Γ ( E 2 , ω ) γ ( ω E ) R 1 δ E [ ( ω E + δ E ) 2 + γ 2 ] [ ( ω E δ E ) 2 + γ 2 ] .
δ φ = ± 2 π 3 , ϑ 1 = ϑ 2 = 1 2 arccos ( 1 3 ) 54.7 ° .
g 1 = 4 χ sin δ φ 2 ( cos δ φ 1 ) χ 2 , g 2 = 4 χ sin δ φ 2 ( cos δ φ 1 ) + 4 csc 2 ϑ 1 + χ 2 ,
absolute maximum : ϑ 66.42 ° , φ = π ;
local maximum : ϑ 0 40.20 ° , φ + 53.73 ° ;
saddle point : ϑ + 21.33 ° , φ = 0 .
E 1 = E δ E , E 2 = E + δ E , c 1 = 1 2 ( 1 , 1 ) , c 2 = 1 2 ( 1 , 1 ) ,
δ E = ( sin ϑ 1 sin ϑ 2 cos δ φ 2 cos ϑ 1 cos ϑ 2 ) η , δ φ = φ 1 φ 2 , η = d 2 / ( ε a 3 ) .
R 1 = R 2 = ρ sin ϑ 1 sin ϑ 2 sin δ φ ,
g 1 , 2 = 4 χ sin ϑ 1 sin ϑ 2 sin δ φ 2 ( sin ϑ 1 sin ϑ 2 cos δ φ + cos ϑ 1 cos ϑ 2 1 ) χ 2 sin 2 ϑ 2 ,
2 cos 2 ϑ 1 cot ϑ 2 = cos δ φ sin 2 ϑ 1 ,
2 cos 2 ϑ 2 cot ϑ 1 = cos δ φ sin 2 ϑ 2 ,
2 cos δ φ cot ϑ 1 cot ϑ 2 = cos 2 δ φ ,
δ φ = ± π 4 , ϑ 1 = ϑ 2 = π 2 .
δ φ = ± 2 π 3 , ϑ 1 , 2 = { 1 2 arccos ( 1 3 ) , π 1 2 arccos ( 1 3 ) } ;
δ φ = ± 3 π 4 , ϑ 1 = ϑ 2 = π 2 .
r 1 = ( 0 , a 3 2 , 0 ) , r 2 = ( a 2 , 0 , 0 ) , r 3 = ( a 2 , 0 , 0 ) ,
d 1 = d ( sin ξ cos φ , sin ξ sin φ , cos ξ ) ,
d 2 = d ( sin ϑ cos ( φ + 2 π / 3 ) , sin ϑ sin ( φ + 2 π / 3 ) , cos ϑ ) ,
d 3 = d ( sin ϑ cos ( φ 2 π / 3 ) , sin ϑ sin ( φ 2 π / 3 ) , cos ϑ ) .
( E A A A E B A B E ) ,
A = ( cos ϑ cos ξ 1 4 ( 6 cos 2 φ 1 ) sin ϑ sin ξ ) η , B = ( 1 3 4 ( 1 + 2 cos 2 φ ) sin 2 ϑ ) η .
E 1 = E B , E 2 = E + δ E , E 3 = E + δ E + , δ E ± = 1 2 ( B ± 8 A 2 + B 2 ) ,
c 1 = 1 2 ( 0 , 1 , 1 ) , c 2 = ( δ E + , A , A ) ( δ E + 2 + 2 A 2 ) 1 / 2 , c 3 = ( δ E , A , A ) ( δ E 2 + 2 A 2 ) 1 / 2 ,
R 1 = ρ 3 2 sin 2 ϑ cos φ ,
R 2 = ρ 3 A 2 δ E + sin ( ϑ + ξ ) A sin 2 ϑ 4 A 2 + B δ E + cos φ ,
R 3 = ρ 3 A 2 δ E sin ( ϑ + ξ ) A sin 2 ϑ 4 A 2 + B δ E cos φ .
( 3 cos 3 φ 2 cos φ ) cos 2 ϑ = 3 ( cos 3 φ + 2 cos φ ) cos 4 ϑ ,
3 ( 3 sin 3 φ + 2 sin φ ) sin 2 ϑ = 4 sin φ .
φ = { 0 , π } , ϑ = { ϑ ± , π ϑ ± } , ϑ ± = 1 2 arccos ( 1 ± 649 36 ) .
20 cos 2 φ = 7 ,
12 sin 2 ϑ = 5 ,
φ = { φ ± , 2 π φ ± } , ϑ = { ϑ 0 , π ϑ 0 } ,
φ ± = arccos ( ± 1 2 7 5 ) , ϑ 0 = arcsin ( 1 2 5 3 ) .
r 1 = ( a 2 , a 2 3 , 0 ) , r 2 = ( a 2 , a 2 3 , 0 ) , r 3 = ( 0 , a 3 , 0 ) , r 4 = ( 0 , 0 , b ) ,
d 1 = d ( cos ( α + 2 π / 3 ) , sin ( α + 2 π / 3 ) , 0 ) , d 2 = d ( cos ( α 2 π / 3 ) , sin ( α 2 π / 3 ) , 0 ) ,
d 3 = d ( cos α , sin α , 0 ) , d 4 = d ( sin ϑ cos φ , sin ϑ sin φ , cos ϑ ) .
( E A A B A E A B A A E B B B B E ) ,
A = 1 4 ( 6 cos 2 α 1 ) η , B = 27 a 4 b sin α ( a 2 + 3 b 2 ) 5 / 2 η .
E 1 = E δ E + , E 2 = E δ E , E 3 = E 4 = E + A , δ E ± = A ± A 2 + 3 B 2 ,
c 1 = ( δ E + , δ E + , δ E + , 3 B ) ( 3 δ E + 2 + 9 B 2 ) 1 / 2 , c 2 = ( δ E , δ E , δ E , 3 B ) ( 3 δ E 2 + 9 B 2 ) 1 / 2 ,
c 3 = 1 2 ( 1 , 0 , 1 , 0 ) , c 4 = 1 2 ( 1 , 1 , 0 , 0 ) ,
R 1 = R 2 = ρ 3 B A 2 + 3 B 2 cos α , R 3 = R 4 = 0 .
( E A A B + C A E A B + C A A E 2 C B + C B + C 2 C E ) ,
B = 9 a 3 ( a 2 6 b 2 ) sin ( α φ ) + 3 a 2 sin ( α + φ ) 4 ( a 2 + 3 b 2 ) 5 / 2 η ,
C = 3 3 a 3 ( a 2 6 b 2 ) cos ( α φ ) 3 a 2 cos ( α + φ ) 4 ( a 2 + 3 b 2 ) 5 / 2 η .
E 1 = E 2 A , E 2 = E + δ E , E 3 = E + A , E 4 = E + δ E + ,
δ E ± = 1 2 ( A ± A 2 + 8 ( B 2 + 3 C 2 ) ) ,
c 1 = 1 3 ( 1 , 1 , 1 , 0 ) ,
c 2 = 1 N + ( B + C , B + C , 2 C , δ E + ) ,
c 3 = 1 6 ( B 2 + 3 C 2 ) ( B 3 C , B + 3 C , 2 B , 0 ) ,
c 4 = 1 N ( B + C , B + C , 2 C , δ E ) ,
R 1 = R 3 = 0 , R 2 = R 4 = σ 3 B cos ( α φ ) 3 C sin ( α φ ) A 2 + 8 ( B 2 + 3 C 2 ) b d 2 .

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