Chiral Optical Properties of Tapered Semiconductor Nanoscrolls

Chiral Optical Properties of Tapered Semiconductor Nanoscrolls Nikita V. Tepliakov,† Anvar S. Baimuratov,† Ilia A. Vovk,† Mikhail Yu. Leonov,† Alexander V. Baranov,† Anatoly V. Fedorov,† and Ivan D. Rukhlenko*,†,‡ †

Center of Information Optical Technologies, ITMO University, Saint Petersburg 197101, Russia Monash University, Clayton Campus, Clayton, Victoria 3800, Australia

‡

ABSTRACT: Large surface-to-volume ratio, one-dimensional quantum confinement, and strong optical activity make chiral nanoscrolls ideal for the detection and sensing of small chiral molecules. Here, we present a simple physical model of chiroptical phenomena in multilayered tapered semiconductor nanoscrolls. Our model is based on a linear transformation of coordinates, which converts nanoscrolls into flat but topologically distorted nanoplatelets whose optical properties can then be treated analytically. As an illustrative application example, we analyze absorption and circular dichroism spectra of CdSe nanoscrolls using an eight-band model of CdSe. We show that the optical activity of the nanoscrolls originates from the chiral distortion of their crystal lattice and determine selection rules for the optically active interband transitions. The results of our study may prove useful for the modeling and design of semiconductor nanoscrolls and nanoscroll-based materials. KEYWORDS: optical activity, circular dichroism, colloidal semiconductor, nanoplatelets, optical spectroscopy, metric tensor, coordinate transformation

C

the nanoscroll dimensions to ensure that the quantum states of charge carriers are resonant.33 Albeit the large number of studies on the formation and characterization of semiconductor nanoscrolls, their chiroptical properties have not been addressed properly so far. It is still unclear what the origin of optical activity in nanoscrolls is, and how the intensity of chiroptical response depends on the nanoscroll’s size and material. In this paper, we comprehensively analyze the optical and chirooptical properties of semiconductor nanoscrolls, answering these questions. Using the original method of coordinate transformation, which virtually unrolls the tapered nanoscroll into the plain nanoplatelet in the curved coordinates, we analytically calculate the absorption and circular dichroism spectra of nanoscrolls. Our simple model may prove useful in interpretation of experimental data on chiral nanoscrolls and can be used to design nanostructures with desirable chiroptical response.

hiral nanomaterials are high-potential base for applications in chemical catalysis,1−6 biomedicine,7−11 and spintronics.12,13 In particular, chiral semiconductor nanocrystals can be used to distinguish between enantiomers of organic molecules by their enantioselective coupling to the molecules of the same handedness.14 Since nanocrystals exhibit an optical response that is much stronger than the optical response of typical organic molecules, the complexes of such nanocrystals coupled to chiral molecules can be used to study the processes of charge, energy, and spin transfer inside the molecules, 15,16 and can be utilized in detection and optomechanical separation of enantiomeric forms of organic drugs.17−21 Nanoscrolls are chiral semiconductor nanocrystals that are formed upon rolling up of nanoplatelets during their synthesis.22−24 Nanoscrolls resemble nanoplatelets in such electronic properties as strong spatial confinement of excitons in one direction, narrow photoluminescence spectra and short luminescence decay time,25,26 strong electrooptical response,27,28 and giant oscillator strengths.29,30 At the same time, nanoscrolls are essentially chiral nanocrystals whose handedness depends on the direction of winding and that are especially attractive for the described applications due to their low density and relatively large surface area.23 Naturally strong optical activity of tapered nanoscrolls can be enhanced even further by doping the nanoscrolls with ionic impurities,31 by introducing screw dislocations inside them,32 or by choosing © 2017 American Chemical Society

RESULTS AND DISCUSSION We describe the shape of a chiral semiconductor nanoscroll via the following system of inequations in cylindrical coordinates {ρ′,φ′,z′}: Received: June 9, 2017 Accepted: July 4, 2017 Published: July 11, 2017 7508

DOI: 10.1021/acsnano.7b04032 ACS Nano 2017, 11, 7508−7515

Article

www.acsnano.org

Article

ACS Nano

Figure 1. Coordinate replacement (eq 2) transforms the tapered nanoscroll with thickness δρ, radius ρ0, and length lz (on the left) into a plain nanoplatelet with dimensions lx = δρ, ly = ρ0φ0, and lz (on the right).

|ρ′ − ρ0 − αρ0 φ′ − βz′| ≤

|φ′| ≤ |z′| ≤

δρ 2

In the nanoplatelet obtained through the coordinate transformation in eq 2, the confined motion of charge carriers is perturbed due to the curvature of space resulting from this tranformation. Mathematically, the modification of the wave functions and energies is described by the perturbation theory. The perturbing potential acting upon an electron or hole is expressed in the terms of metric tensor gij, which is given by

(1a)

φ0 (1b)

2 lz 2

(1c)

Here δρ is the thickness of the rolled nanoplatelet, which is much smaller than its radius ρ0 and length lz, and φ0 is the winding angle, which can exceed 2π if the nanoscroll is multilayered. Parameter α determines the direction of winding and the space between the nanoscroll’s layers, while β alters the nanocrystal radius along the z-axis making the nanoscroll taperedit can be understood from geometrical representation that such a change along the nanocrystal axis is required to remove its mirror symmetry in plane z = 0 and thus make the nanoscroll chiral. Alternatively, β can represent a mechanical strain or ligand field directed along the z-axis. The replacement of coordinates x = ρ′ − ρ0 − αρ0 φ′ − βz′

(2a)

y = ρ0 φ′

(2b)

z = z′

(2c)

gij =

k = 1,2,3

v

⎧ 2 ⎪ cos knvv , for odd nv ⎨ lv ⎪ ⎩ sin knvv , for even nv

∂x′k ∂x′k ∂x′k ∂x′k ≡ ∂x i ∂x j ∂x i ∂x j

(4)

Hereinafter we follow the summation convention for repeated indices, widely used in tensor analysis. By expressing the Cartesian coordinates {x′,y′,z′} in the initial metrics through the coordinates {x,y,z} and substituting them into eq 4, one can write the covariant metric tensor in the form ⎛1 α β ⎞ ⎜ ⎟ αβ ⎟ gij = ⎜ α α 2 + g ⎜⎜ ⎟⎟ 1 + β2⎠ αβ ⎝β

(5)

where 2 ⎛ x + αy + β z ⎞ ⎟⎟ g = ∥gij∥ = ⎜⎜1 + ρ0 ⎝ ⎠

virtually unrolls the nanoscroll, transforming it into a plain nanoplatelet with linear dimensions lx = δρ, ly = ρ0φ0, and lz in Cartesian coordinates {x,y,z} ≡ {x1,x2,x3}, as is demonstrated in Figure 1. The confinement energy of a conductivity electron (e) or hole (h) in the plain nanoplatelet is given by a sum of onedimensional energies, Ee(h) = Ee(h) + Ee(h) + Ee(h) n nx ny nz , where n = (nx,ny,nz) is the set of integer quantum numbers characterizing the quantum state of a charge carrier. Wave function ψe(h) of n either electron or hole is the product of one-dimensional functions34 ψne(h) =

∑

(6)

is the determinant of this metric tensor. Contravariant metric tensor is found as an inverse of eq 5: ⎛1 + β 2 + α 2g −1 −αg −1 −β ⎞ ⎜ ⎟ g ij = ⎜ − αg − 1 g −1 0 ⎟ ⎜ ⎟ ⎜ ⎟ −β 0 1 ⎠ ⎝

(7)

As should be expected, the determinant of this tensor is ∥g ∥ = g−1. Eqs 5−7 are exact and can be significantly simplified by taking into account that the dimensionless parameters α, β, and x/ρ0 are much smaller than unity. The simplification is achieved by expanding these equations in Taylor series up to the first order in α, β, and x. For example, the determinant of the metric tensor can be written in the form g ≈ 1 + 2(x + αy + βz)/ρ0 + O(α2,β2,x2/ρ20). A few significant features of the examined space metrics can be seen from eq 5. For instance, parameter α simultaneously perturbs the motion of charge carriers in the y-direction and couples their motions in directions x and y, whereas parameter β affects the motion in the y- and z-directions while coupling x and z coordinates. If αβ ≠ 0, the shape parameters couple the ij

(3)

where v = x,y,z, and knv = π nv/lv is the wavenumber. These wave functions are good approximation if the confinement energies are small compared to the potential barrier provided by the nanoplatelet’s surface, which is generally different for electrons and holes. If the nanoscroll is multilayered, then the potential barrier on its surface is also high enough to prevent the interlayer tunneling of charge carriers (as is the case where the surface is passivated with small ligands residing between the nanoscroll’s layers). 7509


Article

ACS Nano

Figure 2. (a) Kane parameter 7 inside a plain nanoplatelet illustrating the orientations of unit cells. (b) Left- and (c) right-handed deformations of the crystal lattice, resulting from the coordinate transformation in eq 2. The crystal lattice distortion is described by the i vector field 7̃ = g i17 . Red, blue, and green arrows correspond to different coordinates x inside the nanoplatelets. Figure shows only the fragment of the nanoplatelet’s volume, not related to any specific nanocrystal size.

last pair of coordinates, y and z. This is the only situation, in which chirality can emerge, because it requires all the three Cartesian directions to be interrelated through some perturbation.35 Hence, when evaluating chiroptical properties of semiconductor nanoscrolls, one should keep only the terms that include the product αβ in the final equations, since this is the only term contributing to the nanocrystal optical activity. The perturbing potential in the nontrivial metrics stems directly from the alteration of the kinetic energy operator K̂ = −ℏ2/(2me(h))Δ due to the modification of Laplace operator upon the coordinate transformation. In an arbitrary coordinate system this operator is given by36 Δ = g ij∂i∂j − g ijg kl Γij , l∂k

optical activity. This term reduces all the confinement energies of the carriers by the same value −ℏ2/(2me(h)ρ02). The reduction of the confinement energies results from the interaction of charge carriers with the internal electric field produced by the mechanical strains, which arise when the nanoplatelets are folded into the nanoscrolls. In the experimental studies this can be observed as the red shift of the interband absorption edge

ΔE = −

∂gij ⎞ ∂gil 1 ⎛⎜ ∂glj ⎟⎟ + − 2 ⎜⎝ ∂x i ∂x j ∂x l ⎠

(9)

are the Christoffel symbols of the first kind. Substituting the metric tensor and Christoffel symbols into eq 8, we evaluate Δ and thus find the kinetic energy operator to be ℏ2 K̂ = − (∂ 2x + ∂ 2y + ∂ 2z) + V̂ + V0̂ 2me(h)

(10)

where V̂ is the chiral perturbation potential, in the first order of αβ given by V̂ = αβ

∂ yz ⎞ ℏ2 ⎛ ∂x∂yz ⎜⎜2 − 3 x 3 ⎟⎟ me(h) ⎝ ρ0 ρ0 ⎠

(11)

and V̂ 0 = −ℏ is the achiral term affecting the energy spectra of the confined carriers. When the shape parameters α and β are zero and curvature radius ρ0 is infinitely large, perturbation is absent, and the kinetic energy operator in eq 10 leads to the Schrodinger equation solution given by eq 3. The perturbation operator V̂ is seen to be an odd function of x, y, and z, mixing the quantum states of different parities in all the three spatial directions. This is an inherent property of chiral potentials, which induce optical activity upon transitions in the nanocrystal’s electronic subsystem.33 The last term V̂ 0 in the kinetic energy operator results from the curvature of the nanoscroll and does not contribute to its 2

(12)

where μ = memh/(me + mh) is the reduced exciton mass. This feature allows one to indirectly determine the radius ρ0 of semiconductor nanoscrolls by comparing their spectra with the spectra of unrolled nanoplatelets and measuring the red shift of the absorption or photoluminescence edge. For example, the red shift of photoluminescence spectra of CdSe nanoscrolls was observed in the experimental work.25 Optical activity emerging from perturbing potentials similar to that in eq 11 mostly occurs upon intraband transitions of charge carriers and was thoroughly addressed in our earlier works,37−39 so in the following we focus on interband transitions between the valence and conduction bands of a semiconductor nanoscroll. As was shown,40 the absorption rates of band-to-band transitions are practically unaffected by perturbation of the envelope wave functions of charge carriers, with curvature of space metrics being the only significant factor that alters the transition probabilities. According to Fermi’s golden rule, the rate of interband transition |0⟩ → |n, m⟩ ≡ |n⟩e|m⟩h upon which a conductivity electron in state |n⟩ and hole in state |m⟩ are created, is proportional to the squared matrix element ⟨n, m|Ĥ |0⟩ of lightmatter interaction operator. In our nontrivial space metrics this operator is given by Ĥ = e/(m0c)Aigijpj, where m0 and −e are the free-electron mass and charge, Ai is the vector potential of light wave, and pj is the momentum operator. The matrix element of interband optical transition is thus of the form41

(8)

where Γij , l =

ℏ2 2μρ02

/(me(h)ρ20)x∂x

⟨n , m|Ĥ |0⟩ =

e m0c

∫NC ψneucAig ijpj ψmhuvd3x

(13)

where NC means integration over the nanocrystal volume, and uc and uv are the Bloch functions of the conduction and valence bands, rapidly oscillating in the unit cell. Applying the envelope 7510


Article

ACS Nano

in the square brackets in eq 15 contains the product of initially perpendicular components of the Kane parameter, which indicates that the chiral distortion of the crystal lattice couples the motions of charge carriers in the perpendicular directions. Circular dichroism (CD) spectrum of a tapered nanoscroll is the difference between the transition rates induced by the left and right circular polarizations of light, ΔWn,m(E) = WLn,m(E) − WRn,m(E). In contrast to the total absorption, circular dichroism cannot be treated in the electric dipole approximation, in which both circular polarizations appear to be identical, and the vector potential should be expanded up to the term that is linear in wave vector kj, i.e., Ai = A0(1 + kjxj)qi. The CD signal averaged over the random nanocrystal orientations,35

function multiband approximation we note that the envelope functions are practically constant on the unit cell scale, and the above integral can be evaluated as e ψ eAi g ij 7jψmhd3x m0c NC n e = (n|Ai g ij 7j|m) m0c

⟨n , m|Ĥ |0⟩ =

∫

(14)

where 7j = ∫ uc*pj uvd3x is the Kane parameter, UC stands for UC the integration over the unit cell, and (n|...|m) hereinafter denotes the matrix element evaluated using the envelope functions only. Substitution of 7͠ i = g ij 7j transforms eq 14 into a typical matrix element of interband transition, (n|A 7͠ i|m), where the

ΔWn, m(E) =

i

Kane parameter of the nanocrystal material is the coordinatedependent vector 7͠ i rather than a constant. Physically, this represents a distortion of the nanoplatelet’s crystal lattice, resulting from the formation of the nanoscroll. It is reasonable to assume that the nanoscroll material is uniaxial with the preferential direction x, so that 7x = 7 and 7y = 7z = 7⊥. Choosing vector (7 , 0, 0) as the unit cell “axis”, we can display the curvature of the lattice as is shown in Figure 2, where we plot vector field g i171 = (g 11 , g 21 , g 31)7 in the fragment of the nanoplatelet’s volume. It is seen that the inclination of the unit cell axis is the greatest for negative x, toward the curvature center and where the mechanical strain produced by the rolling is the strongest. At the same time, the deformation of the lattice has rotational pattern, which changes its direction depending on the handedness of the nanoscroll. This makes the crystal lattice of the rolled nanoplatelet chirally distorted, which is the main source of interband optical activity. The total absorption of a semiconductor nanoscroll can be well treated in the electric dipole approximation, in which the vector potential of a propagating wave is assumed to be independent of spatial coordinates: Ai ≈ A0qi, where A0 and qi are the amplitude and polarization of the wave. In an ideal nanoplatelet with undistorted lattice, i.e., with g ij 7j = 7 i , this approximation results in the matrix element of an optical transition of the form ⟨n, m|Ĥ |0⟩ = eA0/(m0c)qi7 i(n|m). Evaluated with the envelope functions from eq 3, the overlap integral becomes (n|m) = δn,m. This implies that the electric dipole (ED) allowed transitions occur only between similar size-quantized states of charge carriers with n = m. In the first order of the small parameters α and β the selection rule for the ED transition is the same, so that the total absorption spectrum of a chiral nanoscroll does not differ significantly from that of a plain nanoplatelet. Substituting into eq 14 the metric tensor expanded up to the first order of perturbation parameters and applying Fermi’s golden rule, one can obtain the following orientation-averaged probability of interband transitions: Wn, m(E) =

(16)

is proportional to the rotatory strength of the transition, R n, m = Im(⟨0|d i|n , m⟩⟨n , m|μi |0⟩)

(17)

where the electric and magnetic dipole moment operators are written in tensor form as di = −exi and μi = −e/(2m0c)εijkxjpk (εijk is the Levi−Civita tensor). It should be noted that the rotatory strength describes the chiroptical response of a semiconductor nanocrystal well if its characteristic size is small compared to the wavelength of the probing light. Despite the fact that the nanoplatelet obtained using the proposed coordinate transformation has large lateral dimensions, the volume of the initial nanoscroll itself is small enough for its optical activity to be reliably described using the rotatory strength approach. Interband matrix elements of the dipole moment operators are calculated classically, e.g., like in the works of Zurita-Sanchez and Novotny.42,43 According to the discussion following Figure 2, we use these results in our study by replacing the Kane parameter 7 i with the coordinate dependent vector field i 7̃ = g ij 7j . The resulting matrix elements are given by ⟨0|d i|n , m⟩ =

ie (m|g in 7n|n) m0ω

⟨n , m|μi |0⟩ = −

e εijk(n|x jg kl 7l |m) 2m0c

(18)

(19)

It is seen from eq 17 that an interband transition is optically active provided that it is both electric dipole (ED) and magnetic dipole (MD) allowed. In an achiral nanocrystal, such as a plain nanoplatelet with an ideal crystal lattice, this condition cannot be satisfied due to the different selection rules for the dipole moment operators. Indeed, as is seen from eqs 18 and 19 with g ij 7j = 7 i , the ED transitions in the ideal nanoplatelet occur between states with quantum numbers n = m, whereas the MD transitions occur between the states of opposite parities with respect to Cartesian coordinate xj (like |nx,ny,nz⟩ → |nx + 1,ny,nz⟩ for transitions along the x-axis). Chiral distortion of the crystal lattice, resulting from the formation of the nanoscroll, mixes the wave functions of different energy bands and thus enables optically active transitions, which involve both the electric and magnetic dipole moments of the nanocrystal. Substituting eqs 18 and 19 into eq 17 and keeping only the term proportional to αβ, we see that the rotatory strength is decomposed into two essentially different terms, one of which,

2 2π ⎛ eA 0 ⎞ ⎜ ⎟ [7 2 + 27 ⊥2 − 4(|α| + |β|)77⊥] 3ℏ ⎝ m0c ⎠

fn, n (E)δn, m

2 4π ⎛ ωA 0 ⎞ ⎜ ⎟ R n, mfn, m (E) 3ℏ ⎝ c ⎠

(15)

Here E is the exciting photon energy and f n,m(E) is the spectral line shape of transition |0⟩ → |n, m⟩. Note that the third term 7511


Article

ACS Nano

Figure 3. (a) Chiral CdSe nanoscroll with thickness 2.2 nm, radius 10 nm, length 40 nm, winding angle 7π/4, α = 0.14, and β = 0.22. (b) Band structure of CdSe along the crystallographic direction [100]: HH, LH, and SO correspond to the heavy, light, and spin−orbit holes, Eg = 1.66 eV is the bangap, and Δ = 0.39 eV is the spin−orbit splitting. Gray arrows indicate the first two electric dipole transitions. (c) Absorption spectrum of the nanoscroll. Electric dipole transitions are labeled according to the valence band, from which they occur, and by the quantum number nx of electron and hole. Dashed curves feature the distinctive absorption bands of the transitions with the same nx. (d) CD spectrum of the two optical isomers of the nanoscroll. Labels of the form SO′1 refer to the first derivatives of the respective lines in the absorption spectrum, and HH2−1 indicates the magnetic dipole transitions |1,ny,nz⟩ → |2,ny,nz⟩ from the band of heavy holes.

R nED ,m =

αβ e 2 [77⊥(n|x 2|n) + 7 (2+)(n|z 2|n)]δn, m ρ0 m02cω

R n, m = (20)

+ lz2(7 (2+)Dnz δn, m − 7 (2−)Bn2zmz )]

is nonzero only for the ED transitions, and the other one, R nMD ,m

(21)

is nonzero for the MD transitions along x or z axis. Here 7 (2±) = 77⊥ ± (7 2 − 7 ⊥2 ). These terms result in two kinds of CD spectra, which differ in sign and positions of the peaks. Using the envelope wave functions from eq 3, we can evaluate the matrix elements of coordinate operators. In particular, the matrix element of x is given by31 (22)

if nx and mx are of different parities; otherwise this matrix element turns zero. Matrix element of x2 is equal to 1 1 1 (n|x 2|n) = − ≡ Dnx 12 lx2 2π 2nx2

(24)

The rotatory strength is seen to comprise two different terms, one of which scales as ∝ l2x while the other one scales as ∝ l2z . We shall refer to these terms as the contributions from the xand z-transitions, respectively. One can see that the chiroptical response of the nanoscroll is determined by its radius ρ0, thickness lx, and length lz, while being completely independent of the nanoplelet’s width ly in the direction of winding. This feature can be attributed to the nanoscrolls geometrical shape, which is close to nanotubeit has only radius, wall thickness, and length. The larger interlayer space α is the worse this approximation becomes, so that the rotatory strength is expected to become dependent on ly when α is comparable to unity. The independence of rotatory strength on ly also implies that the chiroptical response of the nanoscroll is practically unaffected by the rolling up pattern of the nanoplatelet. For example, the nanoplatelets folded from corner to corner would exhibit optical activity that is similar to the activity of the nanoscrolls studied in this work. This freedom in the choice of the winding pattern disappears if the nanoplatelets are made of a fully anisotropic material. In this case, 7y ≠ 7z and the chiroptical response of the nanoscrolls depends on both size ly and the rolling up direction of the nanoplatelets.

αβ e 2 =− [77⊥|(n|x|m)|2 + 7 (2−)|(n|z|m)|2 ] ρ0 m02cω

8n m 1 (n|x|m) = − 2 2 x x 2 2 ≡ Bnxmx lx π (nx − mx )

αβ e 2 [77⊥lx2(Dnx δn, m − Bn2xmx ) ρ0 m02cω

(23)

Similar expressions are obtained for the matrix elements of z and z2. With these results, we can write the rotatory strength of an interband transition in the final form 7512


Article

ACS Nano Even though l2z ≫ l2x, one cannot discard the term in the rotatory strength that corresponds to the x-transitions for the following reason. As this direction is characterized by the strong spatial confinement due to the relatively small nanoplatelet’s thickness lx, the peaks of the ED and MD transitions can be easily distinguished from each other. On the other hand, quantum confinement along z-axis is weak, and the spectral lines corresponding to the ED and MD transitions in the zdirection overlap in the CD spectrum. As the ED and MD transitions contribute to the optical activity with opposite signs and can partially cancel each other, the resulting CD signal from the z-transitions can be comparable in strength with the contribution from the x-transitions or even be weaker. Interestingly, material anisotropy results in an additional CD signal upon the z-transitions, with intensity determined by the difference 7 2 − 7 ⊥2 . It should be noted that anisotropy itself cannot induce optical activity, and the nanoscroll made of an isotropic material can still be optically active. In the case of fully isotropic semiconductor, 7 (2±) = 77⊥ and the contributions from the x- and z-transitions are alike. Now, we can analyze the relation between the absorption and circular dichroism spectra of a chiral nanoscroll. We assume that the nanoscroll shown in Figure 3a is made of CdSe and apply the eight-band effective mass model to describe its band structure.44 This model takes into account both the degeneracy of the valence band at the Γ-point of the Brillouin zone and the nonparabolicity of the energy bands. Optical transitions in CdSe occur between one conduction band and three valence bands: heavy hole (HH), light-hole (LH), and spin−orbit (SO) hole bands, which are shown in Figure 3b. The bandgap energy is Eg = 1.66 eV and the splitting of SO band is Δ = 0.39 eV.30 Heavy hole band, which is isolated from all other bands, is characterized by the parabolic dispersion of energy, EHH = −(γ1 − 2γ2) EK, where γ1 = −0.18 and γ2 = −0.65 are the Luttinger− Kohn parameters of CdSe and EK = ℏ2k2/(2m0) is the kinetic energy of a free electron. Energies of the other three bands are the solutions to the dispersion equation

and LH2 of transitions |2,ny,nz⟩ → |2,ny,nz⟩ from the heavy hole and light hole bands. Figure 3d demonstrates the CD spectrum of the two enantiomeric forms of the nanoscroll. The first exciton peak HH1 is seen as a broad line with the same maximum positions as in the absorption spectrum, while all the other ED allowed transitions manifest themselves in the CD spectrum as the first derivatives of the respective bands in the absorption spectrum (labeled LH′1, SO′1, HH′2, and LH′2), which result from the overlapping of the transitions in the z-direction. As is discussed below eq 24, these ED and MD transitions contribute to the optical activity with opposite signs and have very small energy spacing ΔE. As a consequence, one can write for their overlapping peaks f n,n(E) − f n,m(E + ΔE) ≈ − f ′n,n(E)ΔE, so that each pair of transitions produces the first derivative of the spectral line shape f n,n(E). Between two broad bands of the ED transitions in Figure 3d, one can see a small peak HH2−1 of the opposite CD sign, corresponding to the magnetic dipole transition |1,ny,nz⟩ → |2,ny,nz⟩. Other MD transitions occurring in x-direction are not clearly seen in the spectrum, as they overlap with more pronounced lines of the ED transitions. Yet they manifest themselves as distinct gaps in the pronounced CD signal. For example, the gap at 2.7 eV is produced by the MD transitions |2,ny,nz⟩ → |1,ny,nz⟩ from the heavy and light hole bands. These results make it possible to predesign the chiroptical response of semiconductor nanoscrolls. Since the positions and line shapes of the CD peaks of the x- and z-transitions can be varied independently, one can generate the CD signal of desirable sign and strength in the spectral domain of interest by tailoring the two respective dimensions of the nanoscrolls (lx and lz).

CONCLUSIONS In summary, we have studied the physics behind optical activity of rolled up semiconductor nanoplatelets and proposed a simple analytical model that describes this activity. Upon relaxation of mechanical strains and formation of the tapered nanoscroll, the nanoplatelet acquires chiral deformation of its crystal lattice, which makes possible optically active transitions in the nanoplatelet’s electronic subsystems by mixing the electric and magnetic dipole moments of the nanocrystal. Using an original method of the coordinate transformation we mathematically unrolled the nanoscroll and calculated the distortion of its crystal lattice. We then calculated the total and differential absorption of the nanoscroll and showed that its chiroptical response is independent of the nanoplatelet’s lateral size in the direction of winding. Taking CdSe nanoscroll as an example, we compared the absorption and circular dichroism spectra of this nanocrystal, showing that peaks in its CD spectrum can be easily identified with the lines in the absorption spectrum and vice versa. The results of our study will benefit a wide range of experimentalists and theoreticians interested in optical activity of man-made semiconductor nanostructures and in the methods of its quantification.

{[E + γ1EK + Δ][E + (γ1 + 2γ2)EK ] − 8(γ2EK )2 } × (E − Eg − αEK ) − EpEK (E + 2Δ/3) = Ep(γ1 − 2γ2)EK2

(25)

where Ep = 27 2/m0 = 16.5 eV is the Kane energy and α = −1.54 is the parameter describing the impact of all the bands on the conduction band. Absorption cross section (in the units of cm2) of a single nanoscroll is defined as σ (E ) =

E I

∑ Wn,m(E) n, m

(26)

where I = ω2A20/(4πc) is the intensity of exciting light. The circular dichroism spectrum Δσ(E) is calculated in a similar fashion. The theoretically calculated absorption spectrum of a CdSe nanoscroll shown in Figure 3c is similar to the typical experimental spectra of ordinary semiconductor nanoplatelets.25 It is seen to comprise two exciton peaks HH1 and LH1 corresponding to the lowest transitions from the heavy and light hole bands. It also contains the absorption band SO1 of the electric dipole transitions |1,ny,nz⟩ → |1,ny,nz⟩ from the spin−orbit hole band, and high energy absorption bands HH2

METHODS The interband matrix elements in eqs 13, 18, and 19 were calculated using the envelope wave function approximation by assuming that the full wave functions of electrons (holes) are the products of the form e(h) is the slowly varying (on the length scale of unit ψe(h) n ue(h), where ψn cell) envelope given by eq 3 and ue(h) is the rapidly oscillating Bloch function. The Laplace operator in the curved space was evaluated using the Christoffel symbols of the examined space metrics: 7513


Article

ACS Nano Γ12,2 =

Γ22,2 α

=

Γ32,2 β

=

g ρ0

(15) Nakashima, T.; Kobayashi, Y.; Kawai, T. Optical Activity and Chiral Memory of Thiol-Capped CdTe Nanocrystals. J. Am. Chem. Soc. 2009, 131, 10342−10343. (16) Galoppini, E.; Guo, W.; Zhang, W.; Hoertz, P. G.; Qu, P.; Meyer, G. J. Long-Range Electron Transfer Across MoleculeNanocrystalline Semiconductor Interfaces Using Tripodal Sensitizers. J. Am. Chem. Soc. 2002, 124, 7801−7811. (17) Rukhlenko, I. D.; Tepliakov, N. V.; Baimuratov, A. S.; Andronaki, S. A.; Gun’ko, Y. K.; Baranov, A. V.; Fedorov, A. V. Completely Chiral Optical Force for Enantioseparation. Sci. Rep. 2016, 6, 36884. (18) Freeman, R.; Finder, T.; Bahshi, L.; Willner, I. β-CyclodextrinModified CdSe/ZnS Quantum Dots for Sensing and Chiroselective Analysis. Nano Lett. 2009, 9, 2073−2076. (19) Vovk, I. A.; Baimuratov, A. S.; Zhu, W.; Shalkovskiy, A. G.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Chiral Nanoparticles in Singular Light Fields. Sci. Rep. 2017, 7, 45925. (20) Frasco, M. F.; Chaniotakis, N. Semiconductor Quantum Dots in Chemical Sensors and Biosensors. Sensors 2009, 9, 7266−7286. (21) Andronaki, S. A.; Zhu, W.; Leonov, M. Y.; Shalkovskiy, A. G.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Effect of Extinction on Separation of Nanoparticle Enantiomers With Chiral Optical Forces. IEEE Photonics J. 2017, 9, 1−6. (22) Bouet, C.; Mahler, B.; Nadal, B.; Abecassis, B.; Tessier, M. D.; Ithurria, S.; Xu, X.; Dubertret, B. Two-Dimensional Growth of CdSe Nanocrystals, from Nanoplatelets to Nanosheets. Chem. Mater. 2013, 25, 639−645. (23) Yan, C.; Liu, J.; Liu, F.; Wu, J.; Gao, K.; Xue, D. Tube Formation in Nanoscale Materials. Nanoscale Res. Lett. 2008, 3, 473− 480. (24) Ye, C.; Bando, Y.; Shen, G.; Golberg, D. Formation of Crystalline SrAl2O4 Nanotubes by a Roll-Up and Post-Annealing Approach. Angew. Chem., Int. Ed. 2006, 45, 4922−4926. (25) Vasiliev, R. B.; Sokolikova, M.; Vitukhnovskii, A. G.; Ambrozevich, S.; Selyukov, A.; Lebedev, V. S. Optics of Colloidal Quantum-Confined CdSe Nanoscrolls. Quantum Electron. 2015, 45, 853−857. (26) Dzhagan, V.; Milekhin, A.; Valakh, M. Y.; Pedetti, S.; Tessier, M.; Dubertret, B.; Zahn, D. Morphology-Induced Phonon Spectra of CdSe/CdS Nanoplatelets: Core/Shell vs. Core-Crown. Nanoscale 2016, 8, 17204−17212. (27) Tepliakov, N. V.; Ponomareva, I. O.; Leonov, M. Y.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Field-Induced Broadening of Electroabsorption Spectra of Semiconductor Nanorods and Nanoplatelets. J. Phys. Chem. C 2016, 120, 2379−2385. (28) Tepliakov, N. V.; Leonov, M. Y.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Quantum Theory of Electroabsorption in Semiconductor Nanocrystals. Opt. Express 2016, 24, A52−A57. (29) Achtstein, A. W.; Schliwa, A.; Prudnikau, A.; Hardzei, M.; Artemyev, M. V.; Thomsen, C.; Woggon, U. Electronic Structure and Exciton-Phonon Interaction in Two-Dimensional Colloidal CdSe Nanosheets. Nano Lett. 2012, 12, 3151−3157. (30) Ithurria, S.; Tessier, M.; Mahler, B.; Lobo, R.; Dubertret, B.; Efros, A. L. Colloidal Nanoplatelets With Two-Dimensional Electronic Structure. Nat. Mater. 2011, 10, 936−941. (31) Tepliakov, N. V.; Baimuratov, A. S.; Gun’ko, Y. K.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Engineering Optical Activity of Semiconductor Nanocrystals via Ion Doping. Nanophotonics 2016, 5, 573−578. (32) Mukhina, M. V.; Maslov, V. G.; Baranov, A. V.; Fedorov, A. V.; Orlova, A. O.; Purcell-Milton, F.; Govan, J.; Gun’ko, Y. K. Intrinsic Chirality of CdSe/ZnS Quantum Dots and Quantum Rods. Nano Lett. 2015, 15, 2844−2851. (33) Baimuratov, A. S.; Tepliakov, N. V.; Gun’ko, Y. K.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Mixing of Quantum States: A New Route to Creating Optical Activity. Sci. Rep. 2016, 6, 5. (34) Koch, S. W. Semiconductor Quantum Dots; World Scientific, 1993; Vol. 2.

(27)

where the rest of symbols can be found using the symmetry rule Γij,j = Γji,j = −Γjj,i. All graphs in Figures 2 and 3 were plotted using Wolfram Mathematica 10.

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Ivan D. Rukhlenko: 0000-0001-5585-4220 Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was funded by the Ministry of Education and Science of the Russian Federation under Grant 14.B25.31.0002. The authors also thank the Ministry for its Project 16.8981.2017/8.9 and Scholarships SP-2066.2016.1, SP1975.2016.1 and Grant MD-1294.2017.2 of the President of the Russian Federation for Young Scientists. REFERENCES (1) Wang, Y.; Xu, J.; Wang, Y.; Chen, H. Emerging Chirality in Nanoscience. Chem. Soc. Rev. 2013, 42, 2930−2962. (2) Chiang, W.-H.; Sankaran, R. M. Linking Catalyst Composition to Chirality Distributions of As-Grown Single-Walled Carbon Nanotubes by Tuning NixFe1−x Nanoparticles. Nat. Mater. 2009, 8, 882−886. (3) Kitaev, V. Chiral Nanoscale Building Blocks−from Understanding to Applications. J. Mater. Chem. 2008, 18, 4745−4749. (4) Hu, A.; Yee, G. T.; Lin, W. Magnetically Recoverable Chiral Catalysts Immobilized on Magnetite Nanoparticles for Asymmetric Hydrogenation of Aromatic Ketones. J. Am. Chem. Soc. 2005, 127, 12486−12487. (5) Jansat, S.; Gómez, M.; Philippot, K.; Muller, G.; Guiu, E.; Claver, C.; Castillón, S.; Chaudret, B. A Case for Enantioselective Allylic Alkylation Catalyzed by Palladium Nanoparticles. J. Am. Chem. Soc. 2004, 126, 1592−1593. (6) Tamura, M.; Fujihara, H. Chiral Bisphosphine BINAP-Stabilized Gold and Palladium Nanoparticles With Small Size and Their Palladium Nanoparticle-Catalyzed Asymmetric Reaction. J. Am. Chem. Soc. 2003, 125, 15742−15743. (7) Xia, Y.; Zhou, Y.; Tang, Z. Chiral Inorganic Nanoparticles: Origin, Optical Properties and Bioapplications. Nanoscale 2011, 3, 1374−1382. (8) Sun, C.-Y.; Qin, C.; Wang, C.-G.; Su, Z.-M.; Wang, S.; Wang, X.L.; Yang, G.-S.; Shao, K.-Z.; Lan, Y.-Q.; Wang, E.-B. Chiral Nanoporous Metal-Organic Frameworks With High Porosity As Materials for Drug Delivery. Adv. Mater. 2011, 23, 5629−5632. (9) Govan, J. E.; Jan, E.; Querejeta, A.; Kotov, N. A.; Gun’ko, Y. K. Chiral Luminescent CdS Nano-Tetrapods. Chem. Commun. 2010, 46, 6072−6074. (10) Kommareddy, S.; Amiji, M. Poly(Ethylene Glycol)-Modified Thiolated Gelatin Nanoparticles for Glutathione-Responsive Intracellular DNA Delivery. Nanomedicine 2007, 3, 32−42. (11) Rosi, N. L.; Mirkin, C. A. Nanostructures in Biodiagnostics. Chem. Rev. 2005, 105, 1547−1562. (12) Naaman, R.; Waldeck, D. H. Spintronics and Chirality: Spin Selectivity in Electron Transport Through Chiral Molecules. Annu. Rev. Phys. Chem. 2015, 66, 263−281. (13) Rikken, G. L. A New Twist on Spintronics. Science 2011, 331, 864−865. (14) Elliott, S. D.; Moloney, M. P.; Gun’ko, Y. K. Chiral Shells and Achiral Cores in CdS Quantum Dots. Nano Lett. 2008, 8, 2452−2457. 7514


Article

ACS Nano (35) Berova, N.; Polavarapu, P.; Nakanishi, K.; Woody, R. Comprehensive Chiroptical Spectroscopy: Instrumentation, Methodologies, and Theoretical Simulations; John Wiley & Sons, Inc, 2012; Vol. 1. (36) Landau, L. D.; Lifshitz, E. M. The Classical Theory of Fields; Pergamon, 1971. (37) Baimuratov, A. S.; Tepliakov, N. V.; Gun’Ko, Y. K.; Shalkovskiy, A. G.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Intraband Optical Activity of Semiconductor Nanocrystals. Chirality 2017, 29, 159−166. (38) Rukhlenko, I. D.; Baimuratov, A. S.; Tepliakov, N. V.; Baranov, A. V.; Fedorov, A. V. Shape-Induced Optical Activity of Chiral Nanocrystals. Opt. Lett. 2016, 41, 2438−2441. (39) Tepliakov, N. V.; Baimuratov, A. S.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Optical Activity of Chirally Distorted Nanocrystals. J. Appl. Phys. 2016, 119, 194302. (40) Tepliakov, N. V.; Baimuratov, A. S.; Baranov, A. V.; Fedorov, A. V.; Rukhlenko, I. D. Analytical Study of Optical Activity of ChiralShape Nanocrystals. Proc. SPIE 2017, 1011418−1. (41) Haug, H.; Koch, S. W. Quantum Theory of the Optical and Electronic Properties of Semiconductors; World Scientific Publishing Co. Inc., 2009. (42) Zurita-Sánchez, J. R.; Novotny, L. Multipolar Interband Absorption in a Semiconductor Quantum Dot. I. Electric Quadrupole Enhancement. J. Opt. Soc. Am. B 2002, 19, 1355−1362. (43) Zurita-Sánchez, J. R.; Novotny, L. Multipolar Interband Absorption in a Semiconductor Quantum Dot. II. Magnetic Dipole Enhancement. J. Opt. Soc. Am. B 2002, 19, 2722−2726. (44) Bahder, T. B. Eight-Band k · p Model of Strained Zinc-Blende Crystals. Phys. Rev. B: Condens. Matter Mater. Phys. 1990, 41, 11992.

7515


Chiral Optical Properties of Tapered Semiconductor Nanoscrolls

Recommend Documents