Optical Anisotropy of Topologically Distorted Semiconductor

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Optical Anisotropy of Topologically Distorted Semiconductor Nanocrystals Anvar S. Baimuratov, Tatiana P. Pereziabova, Weiren Zhu, Mikhail Yu. Leonov, Alexander V. Baranov, Anatoly V. Fedorov, and Ivan D. Rukhlenko Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b02203 • Publication Date (Web): 31 Jul 2017 Downloaded from http://pubs.acs.org on July 31, 2017

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Optical Anisotropy of Topologically Distorted Semiconductor Nanocrystals Anvar S. Baimuratov,† Tatiana P. Pereziabova,† Weiren Zhu,∗,‡,¶ Mikhail Yu. Leonov,† Alexander V. Baranov,† Anatoly V. Fedorov,† and Ivan D. Rukhlenko∗,†,¶ †ITMO University, Saint Petersburg 197101, Russia ‡Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China ¶Monash University, Clayton Campus, Victoria 3800, Australia E-mail: [email protected]; [email protected]

Abstract Engineering nanostructured optical materials via the purposeful distortion of their constituent nanocrystals requires the knowledge of how various distortions affect the nanocrystals’ electronic subsystem and its interaction with light. We use the geometric theory of defects in solids to calculate the linear permittivity tensor of semiconductor nanocrystals whose crystal lattice is arbitrarily distorted by imperfections or strains. The result is then employed to systematically analyze the optical properties of nanocrystals with spatial dispersion caused by screw dislocations and Eshelby twists. We demonstrate that Eshelby twists create gyrotropy in nanocrystals made of isotropic semiconductors whereas screw dislocations can produce it only if the nanocrystal material itself is inherently anisotropic. We also show that the dependence of circular dichroism spectrum on the aspect ratio of dislocation-distorted semiconductor nanorods

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allows resonant enhancing their optical activity (at least by a factor of 2) and creating highly optically active nanomaterials.

Keywords: optical activity, nanoparticles, nanoscale materials, circular dichroism, topological defects There is a growing interest in the creation of artificial optical media whose properties are determined by the design of structural elements 1–3 such as semiconductor nanocrystals with discrete and tunable energy spectra. The optical response of semiconductor nanocrystals can nowadays be routinely modified through variations of chemical composition and morphology, 4–6 the degrees of which are only limited by the ingenuity of researchers and the capabilities of fabrication techniques. This approach proved to be especially productive for semiconductor metamaterials. 7–9 Recently, distortions of the crystal lattice itself by various kinds of surface and bulk defects have been also proven to be quite a powerful tool for altering the optical response. 10–12 In this Letter we present an original approach to the calculation of linear permittivity tensor of topologically distorted semiconductor nanocrystals, which is inspired by the Katanaev’s geometric theory of defects 13 and the principles of transformation optics. 14 By regarding the nanocrystal distortion as being produced by the nontrivial metrics of space, our approach enables studying how the linear optical phenomena in nanocrystals are modified by defects, shape irregularities, and strains. 15–17 As an illustrative application example, we systematically analyze, for the first time to the best of our knowledge, anisotropic optical properties of spatially dispersive semiconductor nanocrystals distorted by Eshelby twists and screw dislocations. We show that an ensemble of such nanocrystals represent a gyrotropic medium with tuneable response, which can be resonantly enhanced at certain frequencies by altering the nanocrystal dimensions. We also demonstrate that our theory is suitable not only for the description of linear optical properties, but can also be used to describe nonlinear optical phenomena associated with lattice distortion and the impact of surface ligands on the optical activity of nanocrystals. Our findings and versatile theoretical approach can 2 ACS Paragon Plus Environment

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be used for the design and creation of artificial optical materials and devices, where defects will play key role in achieving new optical functionalities. Our proposed approach is based on the long-wavelength description of the internal crystal structure of topologically distorted semiconductor nanocrystals by a contravariant metric tensor g ij (i, j = 1, 2, 3). The ideal lattice of undistorted nanocrystals corresponds to the trivial or ‘Euclidean’ metrics g ij = δ ij , whereas nontrivial metrics can describe all sorts of surface and bulk defects that change the ideal periodic arrangement of the nanocrystal atoms. The most straightforward ways to change nanocrystal metrics are the introduction of bulk strains, surface transformation, composition modification (the creation of core/shell or alloyed nanostructures), as well as the introduction of linear defects (such as screw or edge dislocations) and point defects (e.g., impurities or vacancies). Using these basic ways to modify the metric tensor, one can change the symmetry of the system, form stationary quantum states as desired for applications, and also significantly change the interaction strength with external fields. Much like in the case of transformation optics, such modifications yield new dielectric functions, which can be used for superlensing and cloaking. 18 The total Hamiltonian of a topologically distorted nanocrystal coupled to an electromagnetic field of vector potential Ai is given by: 19,20

H=

e √ ij e 1 gg pj + Aj + V, √ pi + Ai 2m g c c

(1)

where m and −e are the mass and charge of a free electron, g = det(gij ), c is the speed of light in a vacuum, and pi = −i~∂i is the ith component of the momentum operator. The potential energy V is the sum of the crystal-lattice potential Vcr and the quantum confinement potential Vsh specified by the nanocrystal shape and the environment. In order to find the wave functions and energy spectrum of our system, we treat the light–matter interaction perturbatively. This simplifies Eq. (1) to a sum of the unperturbed


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Hamiltonian

√ pi gg ij pj H0 = + Vcr + Vsh √ 2m g

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(2)

and the interaction term H1 =

e Ai g ij pj . mc

(3)

In this approach, the nontrivial metrics modifies interaction of the distorted nanocrystals with light by changing the kinetic energy of the confined charge carriers and by altering the scalar product of the vector potential and momentum operators. The nonlinear term e2 /(2mc2 )Ai g ij Aj in the sum of Hamiltonians is neglected, but it can be taken into account for high intensities of light fields. The solution to the unperturbed Schrödinger equation H0 Ψ = EΨ is the product of a slowly varying envelope ψµn (r) and a rapidly oscillating Bloch function uµn (r), Ψµn (r) = ψµn (r)uµn (r), where µ is the band number and n is the set of quantum numbers describing the confined motion of charge carriers. This factorization allows excluding the crystal-lattice potential from the Schrödinger equation using the kp perturbation theory, 21 leading to the following equation for envelopes: √ ij pi gg pj √ + Vsh ψµn = Eµn ψµn , 2mµ g

(4)

where mµ is the scalar effective mass, Eµn = E − Eµ is the confinement energy, and Eµ is the energy of the µth band at the Brillouin zone center. If the nanocrystal distortion is small and |g ij − δ ij | 1, then Eq. (4) can be solved perturbatively 22 and the modification of the nanocrystal response comes from two sources. For instance, the optical activity upon interband transitions predominantly comes from the modified Ap operator whereas this activity upon intraband transitions is mainly determined by the modified envelope wave functions themselves (see Supporting Information). Such a differentiation is impossible in strongly distorted nanocrystals, whose wave functions and energy spectrum are significantly modified by the nontrivial metrics. 4 ACS Paragon Plus Environment

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Once the quantum states of the distorted nanocrystal are known, its optical properties can be analyzed using the matrix element of the interaction Hamiltonian H1 . We factorize this element for interband transitions by assuming that the momentum operator pj acts solely on the Bloch functions and operator Ai g ij acts solely on the envelopes. Since the Bloch functions near the Brillouin zone centre are weakly dependent on quantum numbers of the confined motion, we can write the first matrix element in the form Z

(µν)

dr u∗µn (r) pj uνm (r) ≈ pj

.

(5)

UC

The total matrix element is then decomposed as Z

dr Ψ∗µn (r)H1 Ψνm (r) (µν)

where P i = g ij pj

e ≈ mc

Z

(µν)

∗ dr ψµn Ai g ij pj

ψνm =

e hAi P i inm , mc

(6)

and operation h. . .inm is implicitly defined. One can see that the non-

trivial metrics introduces spatial dispersion into the system, making the matrix element of the momentum operator dependent on the position of the unit cell. 23,24 The linear response of nanocrystals to an applied optical field is described by dielectric ij ij ij 25 function εij = εij Since 1 + iε2 , where ε1 and ε2 are the complex-valued Hermitian tensors. ij εij 1 and ε2 are connected through the Kramers–Kronig relations, we shall analyze only tensor

εij 2 , which describes the effects of absorption and circular dichroism (CD). Decomposing according to Eq. (6) the transition matrix elements in the dielectric function of weakly interacting and identically oriented nanocrystals yields: 26

εij 2 (ω, k) = C

X

heikr P i inm heikr P j i∗nm δ(∆E),

(7)

µn,νm

where C = 8N [πe/(mω)]2 , N is the nanocrystal concentration, ω and k are the frequency and wave vector of excitation light, ∆E = Eµn +Eνm +Eg −~ω, and Eg is the bandgap of bulk semiconductor. Notably, the effect of interaction between identically oriented nanocrystals


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depends on the nanocrystal ordering and could potentially lead to the creation of a supercrystal structure or an amorphous body. In particular, a periodic arrangement of chiral quantum dots would create an optically active “quantum-dot supercrystal” with a semiconductor-like band structure. 27 For studying optical properties of the ensemble with randomly oriented nanocrystals, it is most convenient to use Rosenfeld formula 28 (see Supporting Information). Using the Tailor series expansion heikr P i inm ≈ hP i inm + ihkrP i inm + . . ., we find that ij ij l εij 2 (ω, k) ≈ ε2 (ω) + iγl (ω)k ,

(8)

where

εij 2 (ω) = C

X

hP i inm hP j i∗nm δ(∆E),

(9)

µn,νm

γlij (ω) = C

X

hrl P i inm hP j i∗nm − hP i inm hrl P j i∗nm δ(∆E).

(10)

µn,νm

Note that the third-rank tensor γlij (ω) vanishes identically for undistorted nanocrystals made of nongyrotropic semiconductors. The proposed approach is applicable to the analysis of various nonlinear optical phenomena as well. If the excitation intensity is not too high, then it is possible to calculate the rates of multiphoton generation of electron-hole pairs upon two-photon transitions, photoluminescence, and various kinds of light scattering using the perturbation theory in the interaction operators e/(mc)Ai g ij pj and e2 /(2mc2 )Ai g ij Aj (similar to how it is done in Ref. 29, for example). Since a detailed description of nonlinear processes is beyond the scope of this paper, we demonstrate the usefulness of our approach to nonlinear optics by the example of second-order susceptibility. Indeed, the second-order susceptibility describing the interaction of nanocrystals with two optical fields of frequencies ωp and ωq can be written as


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a sum 30

(2)

χijk (ωp + ωq , ωp , ωq ) =

X Cmnl (ωp , ωq )%imn %jnl %klm + Cmnl (ωq , ωp )%imn %knl %jlm mnl

0 0 (ωq , ωp )%imn %klm %jnl , (11) (ωp , ωq )%imn %jlm %knl − Cmnl − Cmnl

where indices m,n, and l contain both the band number and the set of quantum numbers, N ρmm − ρll , 2 2~ (ωmn − ωp − ωq − iγmn )(ωlm − ωp − iγlm ) N ρll − ρnn 0 Cmnl (ωp , ωq ) = 2 , 2~ (ωmn − ωp − ωq − iγmn )(ωnl − ωp − iγln )

Cmnl (ωp , ωq ) =

(12) (13)

ρmm is the probability that the system is in eigenstate m with energy Em , γmn is the damping rate, ωmn = (En − Em )/~, and the modified in nontrivial metrics matrix elements of electric dipole moments are given by

%imn

≡

%iµn,νm

i~e = m(Eµn + Eνm + Eg )

Z

(µν)

∗ dr ψµn g ij pj

ψνm

(14)

for interband transitions and

%imn

≡

%iµn,µm

i~e = mµ (Eµn − Eνm )

Z

∗ dr ψµn g ij pj ψµm

(15)

for intraband transitions. These expressions can be used to calculate the higher-order susceptibilities of topologically distorted nanocrystals. The derived expressions may be applied to nanocrystals of any shapes and compositions. To be more specific, we consider semiconductors whose band structure can be approximated by a single conduction band (c) and three valence bands (vx , vy , vz ) with effective masses mc and mv . 31 When the matrix element of transitions from band vi is polarized along the ith Cartesian axis, that is, when p(cvx ) = (Px , 0, 0), p(cvy ) = (0, Py , 0), and p(cvz ) = (0, 0, Pz ),


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(a)

z

(b)

z

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(c)

z

R b

∆φ

L

Figure 1: Optically active semiconductor nanocrystals with chiral crystal lattices distorted by (a) an Eshelby twist and (b) a positive (b > 0) screw dislocation. Helical nanoribbon in (c) and Eshelby twist are described by the same nontrivial metrics. Helical nanoribbon in (c) and Eshelby twist are described by the same nontrivial metrics; nanocrystal (a) is obtained by twisting a circular cross-section nanorod, while the nanoribbon is obtained by twisting a rectangular nanoplatelet. Eqs. (9) and (10) give the following explicit dependencies on the metric tensor components:

εij 2 (ω) = C

X

δsq δtq |Pq |2 hg is inm hg jt i∗nm δ(∆E),

(16)

n,m,q

γlij (ω) = C

X

δsq δtq |Pq |2 hrl g is inm hg jt i∗nm − hg is inm hrl g jt i∗nm δ(∆E).

(17)

n,m,q

l Introducing a gyration pseudovector Gl according to the equation kγlij sl = eij l G , where

k = nω/c, n is the refractive index, s = k/k, and eij l is Levi-Civita symbol, we write the complex permittivity in the matrix form 

εxx 2 (ω)

z

y



iG (ω) −iG (ω)    yy  . z x εij (ω, G) = (ω) iG (ω) −iG (ω) ε 2   2   iGy (ω) −iGx (ω) εzz (ω) 2

(18)

We next apply the developed formalism to explore the effects that the two most common types of structural chirality — Eshelby twist 32 schematized in Fig. 1(a) and screw dislocation 17,33 sketched in Fig. 1(b) — have on the optical properties of semiconductor nanocrystals. According to elementary theory of dislocations, 34 the image forces acting on a screw dislocation and the associated twist make them form predominantly along the rotational 8 ACS Paragon Plus Environment

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symmetry axis of nanorods or nanowires. 35 Using this fact, we describe an Eshelby twist along the z axis of a nanocylinder by the linear transformation of cylindrical coordinates r = r0 , ϕ = ϕ0 + az 0 , z = z 0 , where a is the angle of relative rotation of two nanocrystal cross sections, which are unit distance apart. The respective metric tensor in cylindrical coordinates (r, ϕ, z) is given by 



0 0  1    2 2 g ij =  0 a + 1/r −a .   0 −a 1

(19)

A screw dislocation along the z axis is generated by the transformation r = r0 , ϕ = ϕ0 , z = z 0 + βϕ0 , which leads to the metrics of the form 



0 0  1   2 g ij =  −β/r2  0 1/r ,   2 0 −β/r 1

(20)

where β = b/(2π) and b is the projection of the Burgers vector on the z axis. Remarkably, Eq. (19) describes not only twisted nanorods but also nanocrystals of other shapes, like the twisted nanoribbon 36 in Fig. 1(c). Accordingly, our theory can be applied to routinely fabricated nanostructures with dislocations and Eshelby twists,, 37,38 as well as to more exotic objects such as nanohelixes, 39 nanosprings, 40 and chirally arranged ensembles of nanocrystals. 41 Owing to the linearity of the considered transformations, cylindrical nanocrystals that become twisted due to the relaxation of the antiplane strain induced by a screw dislocation are described by metrics which is a sum of metrics (19) and (20). While parameters a and b can, in principle, be considered independent of one another, the full relaxation of the antiplane strain always twists nanocrystals in such a way that a = −b/(πR2 ). 32,34 If a cylindrical nanocrystal with an Eshelby twist is made of an isotropic and optically 9 ACS Paragon Plus Environment

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inactive (e.g., zincblende-type) semiconductor, then its interband matrix elements are all alike and expressed through the Kane energy Ep as |Px |2 = |Py |2 = |Pz |2 = mEp /2. The symmetry considerations in this case require that γ312 = γ321 = 0 and γ123 = −γ132 = γ231 = −γ213 = G(ω)/k, reducing Eq. (18) to the form    ij ε2 (ω, s) =   

ε00⊥ (ω)

0

0

ε00⊥ (ω)

iG(ω)sy −iG(ω)sx

y



−iG(ω)s   iG(ω)sx  ,  00 εk (ω)

(21)

where ε00⊥ (ω) = C

mEp X |hg xq inm |2 δ(∆E), 2 n,m,q

(22)

and ε00k (ω) is obtained by replacing g xq with g zq . The obtained result shows that the presence of Eshelby twist in otherwise isotropic semiconductor nanocrystals produces an optically active and axially anisotropic material. For typical values of a, which rarely exceed a few degrees per angstrom, 42 the birefringence induced by the twist is relatively weak, ε00⊥ (ω) ≈ ε00k (ω), and ε00⊥ − ε00k ∝ a2 . According to Eq. (17), the antisymmetric part of tensor εij 2 (ω, s) contains products of the diagonal and off-diagonal components of metrics (19), hence, scaling as the third power of the characteristic twist angle (∝ a3 ). This part of permittivity determines a two-dimensional gyration vector G(ω, s) = G(ω)(sx , sy , 0), whose components are proportional to the CD signal and provide valuable information on the nanocrystal electronic subsystem, including selection rules, transition energies, energy relaxation rates, etc. The comparison of the CD spectra of three nanocrystals in Figs. 2(a)–2(c) shows that the number of optically active interband transitions grows with the aspect ratio R/L of the nanocrystal dimensions. At the same time, following the previously discovered trend for intraband transitions in nanocrystals with screw dislocations, 16 the optical activity of our nanocrystals weakens with their aspect ratio due to reduced lateral confinement.


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Eshelby twist 3

(a)

|x | = |y | = |z | = 

2

b = 2 nm

0

4

CD

-1 2.0

2.1 (b)

2

2.2

Quantum dot

''_I

''_I 0.01

''II

0.00

0 b = – 2 nm

-1 2.0

22x10-5

b = 2 nm

2.1 (e)

6 1

0.02

|z | = 

1

2.3

Nanorod |x | = |y | = 

2

0 b = – 2 nm

3

3

(d)

2

''_I

1

Screw dislocation 44x10-5

Nanorod

CD

2.2

2.3

Quantum dot

''_I

0.01

4

''II

1 2 0

0 0

-1 2.0 3 2

2.1 (c)

2.2 Nanodisk

2.3

0.00

2.0

10 11x10-5

''_I

2.1 (f)

2.2

2.3

Nanodisk 0.01

''_I 5

1

''II

0 -1 2.0

0

2.1 2.2 Photon energy (eV)

2.3

0.00

0

2.0


2.3

Figure 2: Interband spectra ε00⊥ (ω), ε00k (ω), and G(ω) of cylindrical nanorods (R = 5 nm, L = 29 nm), quantum dots (R = 7 nm, L = 15 nm), and nanodisks (R = L = 9 nm) with (a)–(c) Eshelby twist (ε00k ≈ ε00⊥ ) and (d)–(f) screw dislocation for a = −b/(πR2 ) 32 and b = ±2 nm. The values of the CD signals (red and blue curves) are shown on the right axes. Parameter values are: mc = 0.1 m, mv = 0.5 m, n = 3, Eg = 2 eV, and the full width at half maximum of all transitions is 40 meV. In sharp contrast Eshelby twists, screw dislocations make nanocrystals gyrotropic only if they are made of anisotropic (e.g., wurtzite-type) semiconductors. This is due to the specific form of their metric tensor (20), whose diagonal elements are the same as for the undistorted lattice. To gain a clear physical insight into the optical activity emerging in this situation, consider separately nanocrystals made of semiconductors with two types of linear anisotropy: (i) perpendicular to the dislocation axis, with |Px |2 = |Py |2 = mEp /2 and Pz = 0, and (ii) parallel to this axis, with Px = Py = 0 and |Pz |2 = mEp /2. It follows from Eq. (16) that ε00⊥ ε00k in the first case and ε00k ε00⊥ in the second. Moreover, the two types of anisotropy produce nanocrystals with opposite gyration vectors, G(i) = −G(ii) , which scale in proportion to the first power of the Burgers vector projection b. Figures 2(d)–2(f) show spectra ε00⊥ (ω) and G(ω) for nanocrystals with perpendicular anisotropy and the dominant spectrum ε00k (ω)


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for nanocrystals with parallel anisotropy. The material anisotropy is seen to enhance the CD signal by a factor of 1000. At the same time, similar to the case of twisted nanocrystals, the optical activity of dislocation-distorted nanorods is stronger than the optical activities of quantum dots and nanodisks, and the number of CD peaks in the nanorod’s spectrum is the smallest. Now let us use the fact that interband transitions responsible for anisotropies of types (i) and (ii) can occur between different energy bands of the same nanocrystal. Since the band positions can be turned by altering the nanocrystal dimensions, 43 it is possible to create a situation in which a pair of CD peaks produced by the two types of anisotropy overlap, resulting in the enhanced CD signal. We illustrate this possibility by assuming that there are two valence bands of energy E⊥ , which are characterized by transition matrix elements p(cvx ) = (Px , 0, 0) and p(cvy ) = (0, Py , 0), and a band of energy Ek and p(cvz ) = (0, 0, Pz ). Figure 3(a) shows CD spectra produced by the transitions from these three bands to the conduction band of a semiconductor nanorod with the material parameters given in Fig. 2. The detunings of the valence bands from the conduction band energy Ec are Ec − Ek = 2 eV and Ec − E⊥ = 2.11 eV. One can see that the CD peaks at 2.24 and 2.28 eV in both spectra almost coincide, leading to a nearly two-fold enhancement of the total CD signal for energies exceeding 2.15 eV [see Fig. 3(b)]. This example highlights the intriguing prospects of engineering the dispersion of optical response of topologically distorted nanocrystals via the alteration of their spatial dimensions. The relative strength of optical activity of topologically distorted semiconductor nanocrystals is characterized by the ratio of their CD to the total absorption rate, g(ω) = 2G/(ε00⊥ +ε00k ), known as dissymmetry factor . 22 When Eshelby-twisted nanocrystals with a 1/R are made of anisotropic semiconductor with |Px |2 = |Py |2 = |P⊥ |2 and |Pz |2 = |Pk |2 , this factor can be estimated as gnm ∼ akη |hxinm |2 − |hx2 inm | ,

(23)

where n and m are the quantum numbers of electrons and holes and η = 2 |P⊥ |2 − 12 ACS Paragon Plus Environment

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0.004

E

(a) b = 2 nm

0.002

0

-0.002

Eg

Ec − E_I

Ec − EII

isotropic

perpendicular

parallel ×1

×3

g b = 2 nm

-0.004 2.0

×2

CD

2.1

k

-0.004 2.3

2.2

(e)

(b)

1

k

Photon energy

0.02

ε''II

2

0.01

ε''_I

CD peak resonances 0

0 b = 2 nm 2.0

(d)

(c)

b = – 2 nm

0

3

E

0.004

CD

b = – 2 nm


2.3

CD

Figure 3: (a) CD spectra produced by transitions from different energy bands of dislocationdistorted nanorods, (b) respective absorption spectra, and the total CD spectrum of the nanorods. For clarity, the CD spectra in (a) are plotted for screw dislocations of opposite handednesses; panel (a) also shows the total dissymmetry factor of the nanorods. The spectra in Figs. 2(a)–2(c) are produced by nanocrystals made of isotropic semiconductors, with a simple band structure schematized in panel (c), whereas the emergence of optical activity in dislocation-distorted nanorods requires inherently anisotropic materials, whose possible band structure is shown in panel (d). The CD resonances in (a) and (b) are highlighted in panel (e). The values of the CD signals (red and blue curves) are shown on the right axes. |Pk |2 / |P⊥ |2 + |Pk |2 is the linear anisotropy parameter. By taking here into account that a = −b/(πR2 ) in fully relaxed nanocrystals, we see that the dissymmetry factor is proportional to the ratio of chirality scales of matter and light, b/λ (λ = 2π/k). The matrix elements in Eq. (23) represent two kinds of optically active transitions, with different selection rules. Transitions of the first kind, described by nonzero matrix elements hxinm , occur between states with quantum numbers (n, l, p) and (n0 , l ± 1, p), where the principal quantum numbers n0 and n can either differ or coincide. Using the explicit form of hxinlp;n0 ,l±1,p from Ref., 4 the dissymmetry factor of the first kind can be found as (I) gnm

∼ −η

4ξnl ξn0 ,l±1 2 (ξnl − ξn20 ,l±1 )2

2

b , λ

(24)

where ξnl is the nth zero of Bessel function Jl (z). Transitions of the second kind occur between states of electrons and holes with the same


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Ligand impact on the metrics

z h/2

L

0 –h/2

y (a)

x

(b)

Figure 4: (a) Semiconductor nanoplatelet with chiral ligands attached to its surface and (b) xy-twist transformation induced by a single ligand. sets of quantum numbers n, l, and p, for which hx2 inm 6= 0. Evaluation of this matrix element gives (II) gnm

4 1 l2 − 1 b ∼η + 2 . 3 2 ξnl λ

(25) (I)

By taking η = 1, λ = 200 nm, and b = 2 nm, we get gnm ≈ −0.0022 for transitions (II)

(1, 0, p) → (1, 1, p) and gnm ≈ 0.0044 for transitions (1, 0, p) → (1, 0, p). These values are comparable to the typical dissymmetry factors (∼ 10−3 ) of small chiral molecules 28 and agree well with the spectrum g(ω) in Fig. 3(a). As a concluding remark, we would like to note that in many cases the type of chiral ligands determines the shape and topology of the nanocrystals synthesized. 44 If the symmetry of the nanocrystal changes insignificantly in the presence of ligands, then the modification of its electronic structure is predominantly caused by the ligand distortion of the upper layers of the nanocrystal. For instance, by applying the density functional theory to small clusters, Balaz et al. 45 showed that the electronic states of ligands can mix with the states of the inorganic core. This theoretical prediction was recently experimentally confirmed by Markovich, Oron and coworkers, which showed that such mixing can be used to probe the interaction of hole states with the ligands. 46 Besides this, most recently, Tang et al. 47 independently related the origin of the induced chirality to the hybridization of the highest occupied molecular orbitals of CdSe with those of chiral ligands. Using our proposed approach it is not hard to suggest how ligands change the metrics near the nanocrystal surface. Let us consider a semiconductor nanocrystal in the form of a rectangular nanoplatelet of dimensions h × L × L with h L. We also assume that the two largest facets of the nanoplatelet are covered by 14 ACS Paragon Plus Environment

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Nlig of chiral ligands as shown in Fig. 4(a). Every ligand can be modeled as a point surface defect acting on a small area near the ligand attachment point, r0i . Then the sum of local impacts on the surface can be described by the transformation of coordinates

0

x=x +

Nlig X

f (|r0 − r0i |)Tx (r0 , r0i ),

(26)

f (|r0 − r0i |)Ty (r0 , r0i ),

(27)

f (|r0 − r0i |)Tz (r0 , r0i ),

(28)

i=1 Nlig

y = y0 +

X i=1 Nlig

z = z0 +

X i=1

where z0i = ±h/2 is the z component of r0i , f (α) is the rapidly decreasing function of α > 0, and functions Tx(y,z) (r0 , r0i ) are determined by the ligand type. By assuming that each chiral ligand produces a small local twist along the z direction [see Fig. 4(b)] and that f (α) = exp(−α/ζ) we arrive at the transformation

0

x=x −

Nlig X

0

az 0 (y 0 − y0i )e−|r −r0i |/ζ ,

(29)

i=1 Nlig

y = y0 +

X

0

az 0 (x0 − x0i )e−|r −r0i |/ζ ,

(30)

i=1

z = z0,

(31)

Here, as in case of Eshelby twist, parameter a is related to the twist angle and the characteristic length ζ determines the range of the “ligand–surface” interaction. To analyze the optical properties of ligand-distorted nanoplatelets, we use our previous results while taking into account the random position of the ligands using the Monte Carlo method. In conclusion, we have derived, for the first time to the best of our knowledge, a quantummechanical expression for the linear permittivity tensor of topologically distorted semiconductor nanocrystals. The distortion was represented by a nontrivial metrics of space and can therefore describe any kind of defect, shape irregularity, or strain that can occur during 15 ACS Paragon Plus Environment

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the nanocrystal formation. We have systematically analyzed optical anisotropy of cylindrical nanocrystals with screw dislocations and Eshelby twists, which often naturally form in semiconductor nanostructures. Both sources of structural chirality were shown to produce anisotropic absorption and optical activity, whose strengths can be resonantly enhanced by adjusting the nanocrystal dimensions. Eshelby twists were found to be capable of bringing gyrotropy to originally isotropic nanocrystals, whereas screw dislocations can only do so if the nanocrystal material is anisotropic. We hope that this work will stimulate new experimental studies of linear optical phenomena in semiconductor nanocrystals, paving the way to their novel useful applications.

Acknowledgement We thank the Ministry of Education and Science of the Russian Federation (14.B25.31.0002, MD-1294.2017.2, SP-2066.2016.1, SP-1975.2016.1) and Natural Science Foundation of Shanghai (17ZR1414300).

Supporting Information Available (1) Rotatory strengths of interband transitions for weakly distorted nanocrystals; (2) Envelope wave functions of cylindrical nanocrystals; (3) Spectral linewidths of interband transitions.

Notes The authors declare no competing financial interest


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Graphical TOC Entry

Nontrivial metrics gij

Absorption

ε''II

ε''_I

Circular dichroism


Optical Anisotropy of Topologically Distorted Semiconductor

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