Optical Activity and Circular Dichroism of Perovskite Quantum-Dot

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C: Physical Processes in Nanomaterials and Nanostructures

Optical Activity and Circular Dichroism of Perovskite Quantum-Dot Molecules Nikita V. Tepliakov, Ilia A. Vovk, Alexander I. Shlykov, Mikhail Yu. Leonov, Alexander V. Baranov, Anatoly V. Fedorov, and Ivan D. Rukhlenko J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b12041 • Publication Date (Web): 09 Jan 2019 Downloaded from http://pubs.acs.org on January 9, 2019

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The Journal of Physical Chemistry

Optical Activity and Circular Dichroism of Perovskite Quantum-Dot Molecules Nikita V. Tepliakov, Ilia A. Vovk, Alexander I. Shlykov, Mikhail Yu. Leonov, Alexander V. Baranov, Anatoly V. Fedorov, and Ivan D. Rukhlenko∗ Information Optical Technologies Center, ITMO University, Saint Petersburg 197101, Russia E-mail: [email protected]

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Abstract Great prospects of using chiral molecules made of semiconductor quantum dots (QDs) as readjustable elements of optically active materials and devices offer them as a potential material base for chiral nanophotonics. This paper presents a rigorous theoretical framework for the analysis of optical activity of chiral QD molecules composed of an arbitrary number of achiral QDs. We show the power of the proposed framework by analytically calculating the rotatory strengths and dissymmetry factors of quantum transitions to the excited molecular states of a QD molecule made of two perovskite QDs. The analysis of the obtained characteristics leads us to the optimal mutual orientations of the QDs, which correspond to the maximal rotatory strengths and the ultimate values of the dissymmetry factors. The peak rotatory strengths are shown to reach 10−35 erg × cm3 , exceeding the typical rotatory strengths of chiral molecules by four orders of magnitude. The developed theoretical framework equips physicists and engineers with a powerful tool for the modelling and engineering of optically active QD molecules for nanophotonics applications.

1. Introduction The recently discovered giant optical activity of quantum-dot (QD) molecules has rendered them a promising element for advanced photonic devices and an appealing study object of chiral nanophotonics. 1–6 This giant optical activity owes its existence to the delocalization of electronic states and the formation of molecular-like wave functions upon the coupling of QDs into a molecule. 7 A large number of material parameters of QD molecules — including the number, orientations, relative positions, sizes, shapes, and materials of individual QDs — allows one to tune their optical activity over a wide spectral range and as desired for particular applications. By ordering QD molecules into higher-order structures, known as QD supercrystals, 8–12 one can further expand the domain of their useful optical functions. It is not surprising therefore that QD molecules have recently ignited a considerable research 2


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interest, which has already led to the creation of new quantum memories, 13 solar cells, 14 logic gates, 15 and qubits. 16,17 Owing to the delocalization of electronic states, the rotatory strengths of optical transitions in QD molecules can be comparable to those of helix-like QD supercrystals 18 while significantly exceeding the rotatory strengths of single QDs. 19–25 On the other hand, since chiral QDs themselves are much more optically active than small chiral molecules, the rotatory strengths of QD molecules can differ from those of ordinary molecules by five to six orders of magnitude. 26,27 Despite this astonishing fact, which makes QD molecules especially attractive for applications in chiral sensing, 28–31 they still remain somewhat underinvestigated theoretically as compared to chiral semiconductor nanocrystals let alone chiral molecules. While the chiroptical response is currently well understood for individual nanocrystals with various origins of chirality, including screw dislocations, 32 Eshelby twists, 33,34 ionic impurities, 35 and different kinds of chiral shapes, 36–44 a similar theoretical base for QD molecules is meager at best. The rapid progress in the field of chiral nanophotonics and the manifold of potential applications of QD molecules demand the development of advanced theoretical models of this new class of photonic nanostructures. This paper continues our seminal study of chiral QD molecules by generalizing the previously developed theory of their optical activity to the case of multiple, generally degenerate states of individual QDs. We first present a general theoretical framework that enables one to analytically calculate molecular states, rotatory strengths, dissymmetry factors, and CD spectrum of a QD molecule composed on an arbitrary number of QDs. This framework relies on the description of interband transitions to the excited states of isolated QDs by permanent dipole moments, which are related to the respective matrix elements of the momentum operator. We then apply the developed formalism to analyze the optical activity of a QD dimer made of two identical perovskite QDs rotated by a certain angle with respect to one another, and to establish the optimal dimer configurations yielding the strongest chiroptical response. The analytically calculated rotatory strengths and dissymmetry factors lead us to

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simple analytical expressions for the critical angles that determine the optimal configurations for any material parameters of the dimer.

2. Theory 2.1 Molecular states We begin by considering a general case of a chiral molecule composed of N identical per(1)

(2)

(s)

ovskite QDs, each characterized by s excited electronic states |ψn i, |ψn i, . . . , |ψn i of en(0)

ergies E (1) , E (2) , . . . , E (s) , and a ground state |ψn i. Some or even all of these states can be (α) degenerate, with respective energies being alike. The excited states |ψn i (n = 1, 2, . . . , N ; α = 1, 2, . . . , s) of isolated QDs are assumed to be strongly localized and form an orthonormal basis,

(β) hψn(α) |ψm i = δnm δαβ ,

s X N X

|ψn(α) ihψn(α) | = 1.

(1)

α=1 n=1

It is convenient to characterize optical transitions to these states by real dipole moments

(α) ˆ (0) d(α) n = hψn |d|ψn i,

(2)

ˆ = −eˆr is the electric dipole moment, −e is the charge of an electron, |ψn(0) i is the where d (0)

(0)

normalized ground state of the nth QD, and hψn |ψm i = δnm . In the dipole–dipole approximation, the Hamiltonian of the QD molecule is given by the symmetric (N × s) × (N × s) matrix

αβ Vnm

αβ αβ (1 − δnm ), Hnm = E (α) δnm δαβ + Vnm (β) (α) (β) (α) 3 dn · rnm dm · rnm 1 dn · dm − , = ε |rnm |3 |rnm |5

4


(3) (4)

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where ε is the effective permittivity of the QDs and rnm = rn − rm is the distance between the centers of QDs n and m specified by radius vectors rn and rm . The excited molecular states are the linear superpositions of the QD states,

|Ψµ i =

s X N X

(α) (α) Cµn |ψn i,

(5)

α=1 n=1

and the ground molecular state is given by N 1 X (0) |ψn i. |Ψ0 i = √ N n=1

(6)

Much like the states of isolated QDs, the excited molecular states form an orthonormal basis

hΨµ |Ψν i = δµν ,

N ×s X

|Ψµ ihΨµ | = 1,

(7)

µ=1

(α)

which implies that the real coefficients Cµn obey the relations s X N X

(α) Cµn

2

= 1,

N ×s X

(α) (β) Cµn Cµm = δnm δαβ .

(8)

µ=1

α=1 n=1

It follows from Eqs. (3) and (5) that the energies of the excited molecular states, Eµ (α)

(µ = 1, 2, . . . , N × s), and coefficients Cµn are the solutions to the eigensystem problem s X N X

(β) αβ − Eµ δnm δαβ Cµm = 0 (α = 1, . . . , s; n = 1, . . . , N ). Hnm

(9)

β=1 m=1

Since the employed approach does not require the knowledge of the wave functions of isolated QDs, it does not assume the calculation of the wave functions of molecular states as well. However, it is still possible to visualize the molecular states by an effective electric potential created by all the dipoles of the QD molecule in state |Ψµ i. By assuming that the (α)

(α)

weight of dipole dn in state µ is given by coefficient Cµn , we can define the effective electric

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potential in this state as s N (α) 1 X X (α) dn · (r − rn ) Cµn φµ (r) = √ . |r − rn |3 N α=1 n=1

(10)

2.2 Rotatory strengths, dissymmetry factors, and CD spectrum The rotatory strength, Rµ , and dissymmetry factor, gµ , of transition |Ψ0 i → |Ψµ i are defined as 19,45,46

ˆ µ i · hΨµ |m|Ψ ˆ 0 i , gµ = 4Rµ /Dµ , Rµ = Im hΨ0 |d|Ψ 2 ˆ 0 i 2 + hΨµ |m|Ψ ˆ 0 i , Dµ = hΨµ |d|Ψ

(11) (12)

√ ˆ ] is the magnetic moment (m is the free-electron mass) and ˆ = −e ε/(2mc)[ˆr × p where m ˆ = −i~∇ is the momentum operator. It follows immediately from this definition that the p sum of rotatory strengths of all possible transitions from the ground state to the excited molecular states is zero, R1 + R2 + . . . + RN ×s = 0. In order to evaluate the matrix element of the magnetic dipole moment, we use the ˆ ˆ and d: following relation between the matrix elements of operators p

(α) p(α) p|ψn(0) i = mhψn(α) |ˆr˙ |ψn(0) i = imωα0 hψn(α) |ˆr|ψn(0) i = −i n = hψn |ˆ

mE (α) (α) dn . e~

(13)

With this relation, Eqs. (2), (5), and (6) yield:

where σ ˜=

√

s N 1 X X (α) (α) ˆ √ Cµn dn , hΨ0 |d|Ψµ i = N α=1 n=1

(14)

s N i˜ σ X X (α) (α) ˆ 0i = √ hΨµ |m|Ψ Cµn E rn × d(α) , n N α=1 n=1

(15)

(α)

(β)

ε/(2~c) and we have taken into account that hψn |ˆr|ψm i = rn δnm δαβ and that

(α) ˆ (0) hψn |d|ψ m i = 0 for n 6= m.

6


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Substitution of this result into Eqs. (11) and (12) yields: σ ˜ X X (α) (β) (β) (α) Cµn Cµm E dn · rm × d(β) , m N α,β n,m 2 2 σ ˜ 2 X (α) (α) 1 X (α) (α) (α) r n × dn . Dµ = C d + C E N α,n µn n N α,n µn Rµ =

(16) (17)

The knowledge of rotatory strengths and dipole absorption rates (Dµ ) of all molecular transitions allows one to calculate the CD and absorption spectra of the QD molecule. These spectra are given by 37,38

CD(ω) = N A(ω) = N

(4π)2 ω X (γµ /π)Rµ , 3c (~ω − Eµ )2 + γµ2 µ

(4π)2 ω X (γµ /π)Dµ , 2 + γ2 2c (~ω − E ) µ µ µ

(18) (19)

where γµ is the phenomenological linewidth of transition |Ψ0 i → |Ψµ i (2γµ is the full width at half maximum) and N is the concentration of the QD molecules.

3. Results and Discussion As an illustration of the developed theory we consider a QD dimer consisting of two (N = 2) perovskite QDs shown in Fig. 1(a). Suppose that each of the dimer’s QDs has three generally nondegenerate excited states (s = 3), which are characterized by mutually orthogonal dipole (1)

(2)

(3)

moments dn = deX , dn = deY , and dn = deZ directed along the crystallographic axes X, Y , and Z. If the energies of these states are E (1) = E + δ, E (2) = E − δ, and E (3) = E (δ ≥ 0), then δ = 0 corresponds to the cubic perovskite with a degenerate triplet state whereas δ 6= 0 describes the orthorhombic perovskite whose three lowest-energy states are αβ = 0 , the molecular states nondegenerate. In the absence of coupling between the QDs Vnm are doubly degenerate for δ 6= 0 and six-fold degenerate for δ = 0. The dimer is chiral if the orientations of its QDs are different and it lacks a plane or 7



(a)

Page 8 of 22

ɸ1

ɸ2

ɸ3

ɸ4

ɸ5

ɸ6

(b)

Figure 1: (a) QD dimer made of two perovskite QDs and (b) its six molecular states illustrated by isosurfaces φµ (r) = ±d/a2 [see Eq. (10)], which are plotted for ϕ = π/6 and δ = η; red and blue isosurfaces correspond to different signs of φµ (r). QDs in the dimer are separated by distance a, characterized by three orthogonal dipoles each, and rotated by angle ϕ in the xy plane with respect to one another. The dimer is chiral for ϕ 6= 0 and ϕ 6= π. center of symmetry. Let one of the QDs be rotated by angle ϕ in the xy plane with respect to another, as shown in Fig. 1(a). Then the dimer’s Hamiltonian is given by the matrix 



αβ Hnm

0 0 η cos ϕ −η sin ϕ 0   E+δ    0 E−δ 0 η sin ϕ η cos ϕ 0         0 0 E 0 0 −2η  ,  =   η cos ϕ η sin ϕ  0 E + δ 0 0      −η sin ϕ η cos ϕ 0  0 E − δ 0     0 0 −2η 0 0 E

(20)

where η = d2 /(εa3 ) is the characteristic energy of the dipole–dipole coupling and the columns from left to right correspond to states (nm, αβ) = (n1, α1), (n1, α2), (n1, α3), (n2, α1), (n2, α2), and (n2, α3). The structure of this matrix shows that the z-oriented dipoles are decoupled from the four dipoles in the xy plane. By diagonalizing the Hamiltonian of the QD dimer, we find that the energies of its six

8


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μ=1 3δ

μ=2

μ=3

2

Rμ

ρ/5

εμ

2δ

} η = δ/2

μ=4

0

0

0

–δ

–1

(a)

–3δ –�

–� / 2

0 φ

gμ

1

δ

–2δ

} η = 2δ

–ρ / 5 �/2

�

(b) –�

–� / 2

0 φ

(c)

–2 �/2

�

–�

–� / 2

0 φ

�/2

�

Figure 2: (a) Energies, (b) rotatory strengths, and (c) dissymmetry factors (for χ = 0.2) of four optically active molecular states of the QD dimer shown in Fig. 1(a). Solid and dashed curves correspond to η = δ/2 and η = 2δ respectively. molecular states are of the form Eµ = E + εµ , where p δ 2 + η 2 + 2δη cos ϕ, p ε2 = −ε4 = δ 2 + η 2 − 2δη cos ϕ,

(21)

ε5 = −ε6 = 2η.

(23)

ε1 = −ε3 =

(22)

It turns out that molecular states |Ψ5 i and |Ψ6 i are optically inactive (R5 = R6 = 0), with D5 = 0 and D6 = d2 . The rotatory strengths and dissymmetry factors of the rest of molecular states (µ = 1, 2, 3, 4) are given by

Rµ = (−1)µ (ρ/4)(δ/εµ ) sin ϕ, gµ =

(−1)µ 2χδ sin ϕ , (1 + χ2 /2)εµ + η − (−1)µ δ cos ϕ

(24) (25)

where ρ = χd2 and parameter χ = a˜ σ E determines the relative rates of the electric-dipole and magnetic-dipole absorptions. In deriving this result, we have taken into account that δ E and set E (α) ≈ E to simplify the sums in Eqs. (16) and (17). The six molecular-like states of the dimer are illustrated by Figure 1(b), which shows the isosurfaces of the electric potential defined in Eq. (10). While the optically inactive states

9



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have axially symmetric potentials, the active ones are represented by the potentials that are essentially chiral. In agreement with the structure of the Hamiltonian [see Eq. (20)], the figure also shows that the optically inactive states originate due to the interaction of the (3)

(3)

z-oriented dipoles d1 and d2 whereas the active states come from the interaction of the (α)

four in-plane dipoles d1

(β)

and d2 (α, β = 1, 2).

Equations (21)–(25) allow one to draw a number of general conclusions regarding the energy structure and optical properties of the dimer. First of all, the energies, rotatory strengths, and dissymmetry factors have the following symmetry properties: E1 (ϕ ± π) = E2 (ϕ), E3 (ϕ ± π) = E4 (ϕ), R1 (ϕ ± π) = R2 (ϕ), R3 (ϕ ± π) = R4 (ϕ), g1 (ϕ ± π) = g2 (ϕ), g3 (ϕ ± π) = g4 (ϕ). Then, it is easy to see that R1 = −R3 and R2 = −R4 , so that the general sum rule is automatically satisfied (R1 + R2 + R3 + R4 = 0). There is also a summation rule for the dipole absorption rates, which reads: D1 +D2 +. . .+D6 = (3+χ2 )d2 . The maximal splitting of energies E1 (ϕ) and E3 (ϕ) corresponds to the minimal splitting of energies E2 (ϕ) and E4 (ϕ) and vice versa. Such energy splittings are attained for the most strongly interacting parallel dipoles (corresponding to ϕ = 0 and ϕ = ±π), which constitute achiral dimers. These conclusions are illustrated by Fig. 2 for two values of η/δ. It is clear from Eqs. (24) and (25) that the rotatory strengths and dissymmetry factors are the functions of ratio δ/η. In particular, Rµ (δ/η) = (η/δ)Rµ (η/δ). Furthermore, both Rµ and gµ vanish when the energy detuning between the states of isolated QDs is zero (δ = 0), which implies that the chiral dimers made of cubic perovskite are optically inactive. This (1)

(2)

is a consequence of the cancellation of the CD signal produced by dipoles d1 and d2 by (2)

(1)

the signal coming from dipoles d1 and d2 . Further analysis shows that a QD dimer with (1)

(2)

δ = 0 is optically inactive for any orientation of its constituting QDs (i.e. when d2 and d2 are not in the xy plane). The magnitudes of the rotatory strengths are limited by the condition

|Rµ | ≤ Rp ≡ (ρ/4)(δ/η)sign(η−δ) .

10


(26)

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The limiting values Rµ = ±Rp are attained in two different dimers characterized by the angles

µ sign(η−δ) ϕ± , µ = ±cµ arccos (−1) (δ/η)

(27)

where cµ = (−1)δ1,µ +δ4,µ [see Fig. 2(b)]. Hereinafter the plus or minus sign of the critical angle corresponds to the maximum or minimum of the respective function. According to Eq. (26), the peak magnitude of the rotatory strengths (Rp ) can be maximized by adjusting the distance between the QDs. The maximal Rp is attained for δ = η when a = a0 ≡ d2/3 (εδ)−1/3 , provided that such a proximity of the QDs is allowed by their dimensions. In the latter case, the maximal peak magnitude of the rotatory strengths is given by a0 σ ˜ E(d/2)2 . In order to estimate a0 and the maximal rotatory strength of the QD dimer, we need to know the typical value of the dipole moment of interband transitions. This value is p given by the formula d = e(~/Eg ) Ep /(2m), where Eg is the bandgap and Ep is the Kane energy. 47 By taking Eg = 1.67 eV and Ep = 41.6 eV for CsPbI3 , 48 we find that d ≈ 36 D. With this value of d, material parameters ε = 5 and δ = 1 meV, and the exciton transition energy E = 2.4 eV (corresponding to 5 × 5 × 5 nm3 CsPbI3 QDs 48 ), the optimal distance between the perovskite QDs is a0 ≈ 6 nm and the maximal rotatory strength is given by a0 σ ˜ E(d/2)2 ≈ 2.5 × 10−35 erg × cm3 . This value is 104 times larger than the typical rotatory strengths of small chiral molecules, 49–51 100 times larger than the the rotatory strengths of chiral semiconductor nanocuboids, 40 and is comparable to the rotatory strengths of chiral nanoscrolls. 52 The analysis of the parametric dependency of the dissymmetry factor is simplified considerably by the fact that in practice χ2 is much smaller than unity. Indeed, for the above material parameters and E = Eg we find (˜ σ E)−1 ≈ 105 nm, which gives χ2 ∼ 10−3 for a = 6 nm. It is easy to show that the critical angles of dissymmetry factor gµ obey the

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equation

[δ − (−1)µ η cos ϕ] εµ + [η − (−1)µ δ cos ϕ](1 + χ2 /2) = 0.

(28)

Let us first assume that η < δ, in which case only the second bracket in Eq. (28) can be zero. The solution of this equation then yields two critical angles for each molecular state,

ϕ± µ

= ±cµ arccos

p (−1)µ ϑη + cµ δ 2 + ϑ(δ 2 − η 2 ) , (1 + ϑ)δ

(29)

where ϑ = χ2 + χ4 /4. These angles correspond to the peak magnitudes of the respective dissymmetry factors [see the solid curves in Fig. 2(c)]. By taking into account that χ2 1, we find

2 4 g µ ϕ± µ ≈ ±(2 − χ /4) + O(χ ).

(30)

In the opposite case of η ≥ δ, only the first square bracket in Eq. (28) can be zero for µ = 1 or 2 (because ε1 and ε2 are positive). The respective critical angles are given by Eq. (27) and correspond to the maximal magnitudes of their dissymmetry factors

gµ ϕ± µ = ±

2χδ (1 +

χ2 /2)η

+

p

η2 − δ2

.

(31)

These factors are relatively small and limited by the condition |gµ | < 2χ(δ/η) 1. Finally, since ε3 and ε4 are negative, both brackets in Eq. (28) can be zero for µ = 3 or 4 when η ≥ δ. In this instance, the two angles given by Eq. (27) correspond to the minimal magnitude of the dissymmetry factor whereas other four angles

ϕ± µ,k

1+µ

= ±(−1)

arccos

p ϑη + (−1)k δ 2 − ϑ(η 2 − δ 2 ) (k = 1 or 2) (−1)µ (1 + ϑ)δ

(32)

represent the peaks of |gµ | [see the dotted curves in Fig. 2(c)]. Similar to the case of η < δ, we 12


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η = 4δ / 5

2

6 4

1

(a)

η = 5δ / 4

η=δ A(ω)

(b)

|φ|

(c)

CD(ω)

1

�

2

12

0

3

4

2

1

φ0

2

(d)

(e)

(f)

1

� 4

0 2

(g)

6

(h)

(i)

2

1

1

5� 3 4

12

0

–2δ

–δ 0 δ ε = ħω – E

2δ

–2δ

–δ 0 δ ε = ħω – E

2δ

3 4 2 1 –2δ –δ 0 δ 2δ ε = ħω – E

Figure 3: CD and absorption spectra of QD dimers with different η and ϕ for χ = 0.2 and γ = δ/5. The units of CD and absorption are ξ ≡ N (4π)2 ωρ/(3c) and 3ξ/(2χ) respectively. 2 4 have gµ ϕ± µ,k ) ≈ ±(2 − χ /4) + O(χ ) and, hence, the total dissymmetry of optical response

is realized with two (k = 1, 2) different configurations of the dimer for each of the molecular states. The performed analysis gives clear guidelines for the achievement of the ultimate values of the dissymmetry factor of the QD dimers (gµ = ±2) and, consequently, for the practical realization of the total dissymmetry of their optical response. Similar dissymmetry factors have been recently reported for chiral semiconductor nanosprings, 38 optimal configurations of QD molecules, 1 and helical QD supercrystals. 18 They are ten times larger than the typical g-factors of single chiral nanocrystals 21 and by three to four orders of magnitude exceed the

13



Page 14 of 22

dissymmetry factors of small chiral molecules. 53,54 The dependence of the CD signal on the geometric parameters of the QD dimer (a and ϕ) is determined by the interplay of two factors: (i) changes in the magnitudes of the rotatory strengths and (ii) variation of the energy gaps between the molecular states. In particular, for δ 6= 0 all four rotatory strengths scale linearly with a, due to the linear growth of the angular momentum in Eq. (15), and R1 approaches −R2 as a is increased, so that their sum −1 −2 behaves like R1 + R2 ∝ ρ(ε−1 2 − ε1 ) ∝ a . At the same time, the energy gap between these

states decreases like ε1 − ε2 ∝ a−3 . Therefore, as the QDs of the dimer are moved apart, the four CD peaks ρ/ε2 −ρ/ε1 + − 2 2 (ε − ε1 ) + γ (ε − ε2 )2 + γ 2

−ρ/ε1 ρ/ε2 + 2 2 (ε + ε1 ) + γ (ε + ε2 )2 + γ 2

(33)

merge into two peaks of opposite intensities, which decay like ∝ a−2 . Since the CD signal is proportional to the sum of rotatory strengths, each containing sin ϕ as a factor, the dimer is optically inactive for ϕ = 0 and ϕ = π. The CD signal also vanishes for ϕ = ±π/2, due to the degeneracy of molecular states at these angles [E1 = E2 and E3 = E4 , see Fig. 2(a)]. This conclusion is evidenced by the expression for the CD spectrum in the case of η δ: γ CD(ε) ∝ ρη δ

(ε − δ)(ε − 3δ) + γ 2 (ε + δ)(ε + 3δ) + γ 2 − [(ε − δ)2 + γ 2 ]2 [(ε + δ)2 + γ 2 ]2

sin 2ϕ,

(34)

where ε = ~ω − E. Figure 3 shows the typical CD and absorption spectra of perovskite QD dimers for relatively narrow spectral lines, with γ = δ/5. The row panels correspond to six dimer configurations (ϕ = ±π/12, ±π/4, ±5π/12) and the column panels correspond to three dipole–dipole coupling strengths (η/δ = 4/5, 1, 5/4). The four optically active states of the dimer clearly manifest themselves in the CD spectra. As ϕ is increased, peaks 2 and 4 move apart and approach peaks 1 and 3, forming a pair of characteristic shapes in the CD spectrum. The 14


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central CD peaks (2 and 4) are the most pronounced ones regardless of the dimer parameters, which fully agrees with the visible chiralities of the isosurfaces in Fig. 1(b). Since the spectral lines are assumed to be narrow, the peak CD signals are proportional to the rotatory strengths and maximal for η = δ. The dissymmetry of optical activity varies significantly among the four peaks, depending on the dimer configuration and the dipole–dipole coupling strength. For example, in Fig. 3(i) the highest dissymmetry occurs for peaks 4 and 3 whereas in Fig. 3(a) the dissymmetry of peak 3 is strongly suppressed by the excitation of the optically inactive state 6. These examples show that one can find the energy structure and parameters of the dimer by analysing its CD and absorption spectra, much like the tuning of the parameters enables one to engineer the optical activity of the dimer.

4. Conclusions We have developed a general theory of optical activity of chiral QD molecules, which are made of achiral QDs coupled by the dipole–dipole interaction. The theory was used to analytically calculate the rotatory strengths and dissymmetry factors related to the excitation of molecular states of a QD dimer made of perovskite QDs. We found the optimal orientations of the QDs corresponding to the maximal rotatory strengths and the ultimate dissymmetry factors of the dimer. It was also demonstrated that the peak rotatory strengths of the QD dimer can exceed the typical rotatory strengths of chiral molecules by four orders of magnitude, reaching 10−35 erg × cm3 . These results may prove useful to physicists and engineers in view of the very rapid developments in the field of chiral nanophotonics and the host of potential applications of optically active QD molecules.

Acknowledgement Ministry of Education and Science of the Russian Federation (Minobrnauka) (14.Y26.31.0028, 16.8981.2017/8.9, SP-2066.2016.1); Russian Science Foundation (RSF) (18-13-00200). M.Yu.L. 15



gratefully acknowledges 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.

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Optical Activity and Circular Dichroism of Perovskite Quantum-Dot

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