Letter pubs.acs.org/NanoLett
Dislocation-Induced Chirality of Semiconductor Nanocrystals Anvar S. Baimuratov,†,‡ Ivan D. Rukhlenko,*,‡,† Yurii K. Gun’ko,¶,† Alexander V. Baranov,† and Anatoly V. Fedorov† †
ITMO University, 197101 Saint Petersburg, Russia Monash University, Clayton Campus, Victoria 3800, Australia ¶ School of Chemistry and CRANN Institute, Trinity College, Dublin, Dublin 2, Ireland ‡
S Supporting Information *
ABSTRACT: Optical activity is a common natural phenomenon, which occurs in individual molecules, biomolecules, biological species, crystalline solids, liquid crystals, and various nanosized objects, leading to numerous important applications in almost every field of modern science and technology. Because this activity can hardly be altered, creation of artificial active media with controllable optical properties is of paramount importance. Here, for the first time to the best of our knowledge, we theoretically demonstrate that optical activity can be inherent to many semiconductor nanowires, as it is induced by chiral dislocations naturally developing during their growth. By assembling such nanowires in two- or threedimensional periodic lattices, one can create optically active quantum supercrystals whose activity can be varied in many ways owing to the size quantization of the nanowires’ energy spectra. We believe that this research is of particular importance for the future development of semiconducting nanomaterials and their applications in nanotechnology, chemistry, biology, and medicine. KEYWORDS: Circular dichroism, optical activity, intraband absorption, nanoparticles
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the existing materials, creating metamaterials. For example, strong optical activity is exhibited by artificial hyperbolic media, which can act as an extremely broadband polarization splitter for circularly polarized light.15,16 In this Letter, we focus on semiconductor nanowires, the chirality of which stems from their internal structure. This rapidly developing research avenue of nanotechnology shows great promise to come up with optically active materials operating in a wide frequency range.17 In particular, semiconducting nanowires are a fascinating class of nanomaterials, which have been envisaged to find many important applications in photonics, biotechnology, sensing, photovoltaics, and electronics.18−22 The exact cause of internal optical chirality in each particular nanoobject requires a separate investigation. This kind of chirality has two major sources: the surface and the core of the nanoobject.23 For example, an asymmetrical rolling-up of a graphene sheet leads to the chirality in the arrangement of carbon atoms and as a result to the intrinsic chirality of carbon nanotubes.17 Similar to the nanotubes, chiral defects in semiconducting nanowires significantly affect their mechanical, electrical, and optical properties.24−27 Because the majority of nanostructures can be formed with dislocations,26,28−32 we believe that optical activity is likely to be inherent to many kinds of semiconductor nanocrystals. We validate this concept by developing a unified theory of circular dichroism induced by a screw dislocation. Our theory provides a much needed basis
he property of an object not to coincide with its mirror reflection upon translations and rotations, called chirality, is of great importance for biology,1 medicine,2 chemistry,3 and physics.4 Chiral molecules and nanosized objects can be differentiated from each other by their absorbance of circularly polarized light or circular dichroism (CD). Molecular chemistry has accumulated a great deal of experimental data on chiral structures while a significant progress has been made in the theoretical account for their properties.5 The first quantummechanical treatment of molecular chirality was given by Rosenfeld, who addressed the interaction between a molecule and an electromagnetic field semiclassically, in the second-order of the time-dependent perturbation theory.6 By applying Rosenfeld’s approach to a quantum oscillator, Condon et al. showed that an achiral system perturbed by a dissymmetric potential becomes optically active.7 Much later, it was shown using a toy model of an electron on an infinite helix8 that the approximation of Rosenfeld is inadequate for the description of large-scale molecular aggregates. Having cited three of the pivotal theoretical works on chirality as an example, we refer the reader interested in detailed theory and simulation methods of chiral molecular systems to the special-purpose monograph.5 Most of chiral molecules exhibit optical activity in the ultraviolet range. To shift the CD activity range to longer wavelengths and simultaneously enhance the chiral properties of the molecules, they can be coupled to metallic nanoparticles.9−11 It is also possible to shift optical activity to visible and infrared frequencies via the arrangement of achiral nanocrystals in pyramids, helices, and other chiral forms.12−14 Yet another way to produce optical activity is to nanostructure © XXXX American Chemical Society
Received: November 14, 2014 Revised: January 22, 2015
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DOI: 10.1021/nl504369x Nano Lett. XXXX, XXX, XXX−XXX
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Figure 1. Band spectrum of electrons in quantum nanowires (a) without and (b) with a right-hand screw dislocation b = 2a. It was assumed that R = 20a and the energy is in the units of ℏ2/(2ma2). (c) Schematic of electron transitions illustrating the absence of circular dichroism upon intraband dipole absorption of circularly polarized light by the dislocation−distorted nanowire.
dislocation-induced elastic deformation of the nanocrystal, the pure geometrical travel of electrons in the topologically distorted lattice, and the nanocrystal’s confinement, respectively. This equation neglects the Eshelby twist, which is the nanocrystal’s torque around the screw dislocation’s axis due to the stress and strain induced by the dislocation.40 If the screw dislocation is oriented along the z-axis, then its potentials in cylindrical coordinates are U1 = −2b2/(2πar)2 and U2 = b/(πr2) ∂φ∂z, where a is the undistorted-lattice constant and b is the z component of the Burgers vector. The symmetry and height of the confining potential are determined by the shape, size, and material of the nanocrystal, as well as by the host material. As we mentioned earlier, optical activity can originate from the dissymmetry of V caused by the nanocrystal shape but, as long as such a dissymmetry is absent, the exact form of V is insignificant for the emergence of the nanocrystal’s chirality. With this in mind, we take V as an infinitely large potential barrier at r = R to model a cylindrical nanowire of an impenetrable surface. We then find the wave function obeying eq 1 inside the nanowire to be given by41
for a deeper understanding of the optical activity of various kinds of dislocation−distorted quantum nanostructures, which are nowadays routinely fabricated with various techniques.33,34 It assumes that the nanocrystal chirality comes from the dissymmetry of the dislocation-induced potential affecting the confined electrons. We go beyond the Rosenfeld approximation and take into complete account the retardation of the electric field over the volume occupied by the nanocrystal. We show that the presence of a screw dislocation in a cylindrical nanowire inevitably leads to the circular dichroism of its absorption bands. This result suggests dislocation-enabled semiconductor nanocrystals as versatile building blocks for the creation of next-generation optically active materials with controllable properties. Screw Dislocations in Semiconductor Nanocrystals. The topological distortion of an ideal nanocrystal lattice by a screw dislocation breaks the lattice symmetry and alters the electronic subsystem of the nanocrystal. Being mostly pronounced on the length scale much larger than the atomic spacing, the distortion predominantly affects the long-wavelength quantum states of the nanocrystal. These states are described by the products of a dislocation-modified envelope wave function, ψ(r), and a Bloch amplitude, u(r). The modification of envelopes gives rise to the optical activity upon both interband35,36 and intraband transitions whereas the modification of amplitudes is usually neglected. To address the effect of screw dislocation on the nanocrystal chirality, we focus on the intraband transitions of electrons in semiconductor nanocrystals with a simple-cubic lattice. Using the two-band model of semiconductor and describing the dislocation within the frame of the differential-geometric approach based on the continuum theory of defects in solids,37 we require the longwavelength quantum states of the confined electrons to obey the Schrödinger equation38,39 (Δ + U1 + U2 + V )ψ = −εψ
ψnlk =
1 Jλ (ξnλr /R ) i(kz + lφ) e 2 πR Jλ + 1(ξnλ)
(2)
where ξnλ is the nth (n = 1, 2, 3,...) zero of the Bessel function Jλ(u), l = 0, ±1, ±2,... is the angular momentum, λ = [l2 + blk/π + 2b2/(2πa)2]1/2, and k is the wavenumber. The product blk in the Bessel function order tells us that the screw dislocation couples the radial, angular, and axial motions of electrons while discriminating between the quantum states of opposite handednesses. These features are reflected in the nanowire’s energy spectrum, which comprises the size2 quantized subbands εnl(k) = k2 + R−2ξnλ(l,k) . Ten subbands of an undistorted nanowire and a nanowire with a right-hand screw dislocation b = 2a are plotted in Figure 1a,b. We see that the dislocation blue shifts all the subbands with respect to their positions in the dislocation-free nanowire and removes the double degeneracy of the subbands with nonzero angular momentum. Note that the zero-momentum subbands are
(1)
where Δ is the Laplacian, ε = 2mE/ℏ2 is the reduced electron energy, and the potentials U1, U2, and V represent the B
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Figure 2. Theoretically simulated CD spectra of ZnS nanowire with a right-hand screw dislocation b = a and R = 20a for three kinds of intraband transitions. Insets show variations of angular momentum upon the transitions of each kind.
parabolas with minima at the Brillouin zone center, whereas the pairs of the dislocation-split subbands are nonparabolic and have equal minima located symmetrically with respect to the origin. We unambiguously demonstrate below that the dislocation-induced degeneracy removal of the electronic subsystem is the cause of circular dichroism in topologically distorted nanowires. Circular Dichroism Theory. Circular dichroism (CD) of a semiconductor nanowire is the difference in its absorption efficiencies of the left-hand-polarized and right-hand-polarized light.5 The dipole absorption efficiency is often calculated by neglecting the finiteness of the photon momentum and the associated inhomogeneity of the electric field over the nanowire volume. The conservation of electron’s wavenumber k, following from this approximation, essentially kills the CD of nanowires with screw dislocations. Indeed, if a circularly polarized photon excites an electron from subband (n, l) to subband (n′, l′), then a photon of opposite handedness and same energy excites with equal probability an electron of opposite wavenumber from subband (n,−l) to subband (n′, −l′), resulting in essentially zero CD. This statement is illustrated by Figure 1c, showing the two kinds of the dipoleallowed transitions between the subbands of zero and unity momenta. Note that CD is also absent in the less crude approximation in which the absorption efficiency of a monodisperse ensemble of randomly oriented nanowires is calculated by taking into account the electric quadrupole and magnetic dipole contributions. The measure of optical activity of such an ensemble upon a transition |nlk⟩ → |n′l′k′⟩ is the rotary transition strength defined as5,6
Im(⟨nlk|r|n′l′k′⟩·⟨n′l′k′|r × p|nlk⟩)
(3)
When the matrix elements in this scalar product are evaluated using the wave functions given in eq 2, the rotary strength vanishes due to different selection rules of l for the coordinate and momentum operators. The proper way to calculate CD of a nanowire whose length is not small compared to the wavelength of light is to consider the exact spatial variation of the electric field over the nanowire’s volume.8 To do so, we assume that excitation light propagates along the nanowire and describe its interaction with the confined electrons by the Hamiltonian42 ⎛∂ i ∂ ⎞ H ± = H0ei(qz ∓ φ)⎜ ∓ ⎟ r ∂φ ⎠ ⎝ ∂r
(4)
in which H0 is the electron−photon interaction strength, q is the z component of the photon wave vector, and the upper and lower signs correspond to the right-hand and left-hand polarized light, respectively. It is convenient to investigate CD upon intraband transitions using a pump−probe scheme. We shall therefore assume an experiment in which a linearly polarized optical pump generates electrons in a subband of zero momentum or in a pair of subbands (n, ±l), after which the electrons are excited to the higher-energy subbands by a weak circularly polarized probe. The subbands of opposite momenta are considered equally populated with total electron density N determined by the pump power. The distribution of electrons over wave numbers in subband (n, l) at temperature T is given by the Fermi function C
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Figure 3. Theoretically simulated CD spectra upon intraband transitions (1, ±2) → (1, ±3) in ZnS nanowires with (a) right-hand (solid curves) and left-hand (dashed curves) screw dislocations, R = 20 a and Na = 0.01 at T = 10 K; (b) b = a and Na = 0.01 at T = 10 K; (c) R = 20a, b = a and Na = 0.01; (d) R = 20a and b = a at T = 10 K. Dotted lines in (a) and (b) are the absorption spectra of linearly polarized light. Inset in (d) shows the relevant electronic subbands and distributions f nl(k) for three concentrations of electrons. Solid red spectra are the same in all cases.
⎡ ⎛ ε (k ) − ε ⎤−1 ⎞ nl 0 fnl (k) = 2⎢exp⎜ − μ⎟ + 1⎥ ⎠ κT ⎣ ⎝ ⎦
momentum subbands to subbands with a smaller momentum magnitude, |l′| = |l| − 1. All spectra have the shape of a few deviation peaks from the zero level and characteristic widths ranging from 0.03 to 0.3 meV. We see that the CD upon transitions of the first kind, illustrated by the spectra in Figure 2a−c, grows smaller as the principal quantum number n of the final electron state increases. This is a reflection of the general fact that the overlap integral of the radial part of the envelope wave function and its derivative decreases with |n′ − n|. Note that the spectral features observed are separated by large intervals of an almost zero CD signal, for example, from 12 to 62 meV and from 63 to 144 meV for intraband transitions from subband (1, 0). Transitions of the second kind, represented by the spectra in Figure 2d−f, show CD signals that are about hundred times stronger than those resulting from transitions of the first kind, whereas the signals generated upon transitions of the third kind in Figure 2g−i are only ten times stronger. Such a large variation of the CD signal strength is a consequence of the constructive and destructive interferences of probability amplitudes describing transitions due to the radial and angular derivatives in the electron−photon Hamiltonian. The constructive interference occurs for transitions of the second kind such that left-hand polarized light increases the positive angular momentum and right-hand polarized light decreases the negative momentum, whereas the destructive interference takes place for transitions of the third kind upon which the negative momentum is increased by left-hand polarized light and the positive momentum is reduced by right-hand polarized light (see insets in Figure 2). A notable difference between the CD upon transitions of the first group and those coming from transitions of the second and third groups is the opposite trends in their strength variations
(5)
where ε0 is the minimum energy of subband (n, l), μ is the effective chemical potential, κ = 2mkB/ℏ2, and kB is the Boltzmann constant. By neglecting depletion of the pumpexcited subbands and assuming that the rest of the subbands remain vacant due to the low probe power and fast intraband relaxation, we can write the intraband absorption rate of circularly polarized light using the Fermi golden rule43 as W± ∝
∫ dk ∫ dk′ ∑ ∑ fnl (k)
n′l′k′ H ± nlk 2 δ(ω − ωnlk ; n ′ k ′ l ′)
±l n ′ l ′
(6)
where ωnlk;n′k′l′ = ℏ[εn′l′(k′) − εnl(k)]/2m. This expression involves summations and integration over all the final states |n′l′k′⟩ shifted from the initial state |nlk⟩ by the photon energy ℏω, summation over the pump-excited subbands with opposite angular momenta and averaging over the energy distribution of the excited electrons. As expected, evaluation of this expression with the envelope wave functions of electrons in a dislocationfree nanowire (with b = 0) gives W+ = W−. Details of the analytical calculation of CD, ΔW = W− − W+, in the presence of a screw dislocation can be found in Supporting Information. Results and Discussion. Figure 2 shows CD spectra upon different intraband transitions in ZnS nanowire with a righthand screw dislocation b = a, R = 20a and Na = 0.01 at T = 10 K (for parameters of ZnS see Supporting Information). It is seen that all transitions can be classified into three groups according to the strength of their CD: (i) from a zeromomentum subband to subbands with l = ±1; (ii) from a pair of nonzero-momentum subbands to subbands with a larger momentum magnitude, |l′| = |l| + 1; and (iii) from nonzeroD
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is of the order of unity near the intraband resonance and equal unity out of the resonance. Such large g-factors are indicative of an extremely pronounced CD of the nanowires, much more pronounced than the molecular CD, which is significantly weaker than absorption. This is a consequence of infinite nanowire length and relatively simple treatment of intraband transitions. A measurement of CD and g-factor of real semiconducting nanowires with screw dislocations is still pending. Concluding Remarks. We have demonstrated that optical chirality is induced by a screw dislocation in a cylindrical nanowire of length much larger than the diameter. This conclusion still stands for true nanocrystals, with all spatial dimensions in the nanometer scale and fully discrete spectra. Although the result that CD originates from screw dislocations is general in nature, it is the easiest to demonstrate this fact for the special case of an impenetrable circular nanorod. This can be done by analytically solving eq 1 without the kinetic part of (0) the dislocation potential, (Δ + U1 + V)ψ(0) j = −εjψj and then treating the effect of U2 perturbatively. In the first order of perturbation theory, the wave functions of electrons inside the nanorod are given by
with probe frequency. The CD signal is seen to weaken with the frequency of intraband transitions 0 → ± 1, and grow stronger with the frequency of the other transitions. Also noteworthy is that the right-hand screw dislocation results in negative CD peaks upon transitions of the first and second kinds, and in positive peaks upon transitions of the third kind between the states of nonzero momenta. Hence, the maximum optical chirality occurs upon transitions between the electronic subbands with minimal |n′ − n| and |l′| > |l|. The nature of the modification of the CD spectrum is illustrated by Figure 3. In panel a, we see that an increase of the dislocation strength (Burgers vector magnitude) red shifts, broadens, and reduces the CD spectrum whereas a change of the dislocation handedness inverts the spectrum. Because neither dislocation strength nor handedness change the nanowire volume, the integral absorption is preserved. Panel b of the figure shows that the CD spectrum narrows with the nanowire’s radius and red shifts according to the well-known 1/ R2 law peculiar to the size-quantized energy spectra of semiconductor nanowires.41 In each case, the strongest CD dip (peak) is nearly at the position of the absorption resonance of linearly polarized light shown by the dotted curve. Figure 3c,d shows how the CD spectrum changes with the nanowire temperature and electron density, which are responsible for the energy distribution of electrons in the initial subband. We see from panel c that the spectrum broadens with temperature due to the rise in the number of electrons with large wave numbers. The room-temperature spectrum corresponds to the situation where the thermal energy (about 26 meV) exceeds the energy separation of the subbands and, thus, thermal excitation of the subbands needs to be taken into account. Considering such excitation would reduce the intensity and smear the peaks on the low-energy side of the CD spectrum (slightly above and below 26 meV) while almost not affecting its high-energy side. Panel d shows that higher densities of electrons generated by the pump yield stronger CD signal while only slightly affecting its spectral shape and position (note the quantitative difference between the three spectra, two of which are magnified by factors 2 and 100). The qualitative change of the spectral shape becomes noticeable only when the energy distribution turns step-like, as shown in the inset for Na = 1. According to Figure 3a,b, the CD of our nanowires is comparable in strength to the intraband absorption rate. The ratio of the two, known as the CD g-factor5 gCD = (W− − W+)/ (W− + W+), is plotted in Figure 4. One can see that the g-factor
ψj =
ψj(0)
+
∑ i≠j
⟨ψi(0) U2 ψj(0)⟩ εj − εi
(7)
where the subscripts i and j denote the sets of quantum numbers of electrons. By expanding the exponential in the electron photon interaction Hamiltonian as eiqz ≈ 1 + iqz and using the perturbed wave functions in eq 5, it is possible to calculate the strength of CD for different intraband transitions inside the nanorod. The dislocation-induced optical chirality of semiconductor nanorods will be addressed in detail in our future publications elsewhere. The CD spectrum of interband transitions will be the subject of detailed theoretical study elsewhere. Such a study may require more complex models of semiconductor bands, considering the anisotropic nature of the effective mass of holes, as well as taking into account the modification of the Bloch amplitudes. Our study suggests that most of the previously reported semiconducting nanowires and their complexes18,44−46 could have exhibited pronounced optical activity. This activity was not observable with the standard methods of the CD spectroscopy, as the nanowires of opposite handednesses form racemic mixtures of equal amounts of enantiomers, due to equal probabilities of formation of the left-handed and righthanded chiral features. Our study is, thus, of particular importance for the future development of semiconducting nanocrystals for photonics and nanobiotechnological applications. We believe that the CD-active semiconducting nanowires can potentially serve as new emitters of circularly polarized light, chemo-biosensors, and nanoprobes for molecular recognition of various chiral molecules, including biopharmaceutical products and common drugs. Our findings will also prove useful in a highly topical area of nanotoxicology, as most of biomolecules and biological species are chiral, interacting differently with right-handed and left-handed nanocrystals. These findings should thus be taken into account in the future research on the toxicity of nanostructures.
Figure 4. Theoretically calculated CD g-factor upon transitions (1, ±2) → (1, ±3) in ZnS nanowires for the same material parameters as in Figure 3a. E
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ASSOCIATED CONTENT
S Supporting Information *
Quantum states of semiconductor nanowire with a screw dislocation; electron−photon interaction; intraband transition rates; circular dichroism strength; and material parameters. This material is available free of charge via the Internet at http:// pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank the Science Foundation Ireland for Grant SFI 12/IA/1300, the Ministry of Education and Science of the Russian Federation for Grants 3.17.2014/K and 14.B25.31.0002, and the scholarship of the President of the Russian Federation for young scientists and graduate students (2013−2015). The work of I.D.R. is funded by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055. I.D.R. and A.S.B. also gratefully acknowledge the Monash Researcher Accelerator Program and the Dynasty Foundation Support Program for Physicists.
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DOI: 10.1021/nl504369x Nano Lett. XXXX, XXX, XXX−XXX
Dislocation-Induced Chirality of Semiconductor Nanocrystals Anvar S. Baimuratov,†,‡ Ivan D. Rukhlenko,*,‡,† Yurii K. Gun’ko,¶,† Alexander V. Baranov,† and Anatoly V. Fedorov† †
ITMO University, 197101 Saint Petersburg, Russia Monash University, Clayton Campus, Victoria 3800, Australia ¶ School of Chemistry and CRANN Institute, Trinity College, Dublin, Dublin 2, Ireland ‡
S Supporting Information *
ABSTRACT: Optical activity is a common natural phenomenon, which occurs in individual molecules, biomolecules, biological species, crystalline solids, liquid crystals, and various nanosized objects, leading to numerous important applications in almost every field of modern science and technology. Because this activity can hardly be altered, creation of artificial active media with controllable optical properties is of paramount importance. Here, for the first time to the best of our knowledge, we theoretically demonstrate that optical activity can be inherent to many semiconductor nanowires, as it is induced by chiral dislocations naturally developing during their growth. By assembling such nanowires in two- or threedimensional periodic lattices, one can create optically active quantum supercrystals whose activity can be varied in many ways owing to the size quantization of the nanowires’ energy spectra. We believe that this research is of particular importance for the future development of semiconducting nanomaterials and their applications in nanotechnology, chemistry, biology, and medicine. KEYWORDS: Circular dichroism, optical activity, intraband absorption, nanoparticles
T
the existing materials, creating metamaterials. For example, strong optical activity is exhibited by artificial hyperbolic media, which can act as an extremely broadband polarization splitter for circularly polarized light.15,16 In this Letter, we focus on semiconductor nanowires, the chirality of which stems from their internal structure. This rapidly developing research avenue of nanotechnology shows great promise to come up with optically active materials operating in a wide frequency range.17 In particular, semiconducting nanowires are a fascinating class of nanomaterials, which have been envisaged to find many important applications in photonics, biotechnology, sensing, photovoltaics, and electronics.18−22 The exact cause of internal optical chirality in each particular nanoobject requires a separate investigation. This kind of chirality has two major sources: the surface and the core of the nanoobject.23 For example, an asymmetrical rolling-up of a graphene sheet leads to the chirality in the arrangement of carbon atoms and as a result to the intrinsic chirality of carbon nanotubes.17 Similar to the nanotubes, chiral defects in semiconducting nanowires significantly affect their mechanical, electrical, and optical properties.24−27 Because the majority of nanostructures can be formed with dislocations,26,28−32 we believe that optical activity is likely to be inherent to many kinds of semiconductor nanocrystals. We validate this concept by developing a unified theory of circular dichroism induced by a screw dislocation. Our theory provides a much needed basis
he property of an object not to coincide with its mirror reflection upon translations and rotations, called chirality, is of great importance for biology,1 medicine,2 chemistry,3 and physics.4 Chiral molecules and nanosized objects can be differentiated from each other by their absorbance of circularly polarized light or circular dichroism (CD). Molecular chemistry has accumulated a great deal of experimental data on chiral structures while a significant progress has been made in the theoretical account for their properties.5 The first quantummechanical treatment of molecular chirality was given by Rosenfeld, who addressed the interaction between a molecule and an electromagnetic field semiclassically, in the second-order of the time-dependent perturbation theory.6 By applying Rosenfeld’s approach to a quantum oscillator, Condon et al. showed that an achiral system perturbed by a dissymmetric potential becomes optically active.7 Much later, it was shown using a toy model of an electron on an infinite helix8 that the approximation of Rosenfeld is inadequate for the description of large-scale molecular aggregates. Having cited three of the pivotal theoretical works on chirality as an example, we refer the reader interested in detailed theory and simulation methods of chiral molecular systems to the special-purpose monograph.5 Most of chiral molecules exhibit optical activity in the ultraviolet range. To shift the CD activity range to longer wavelengths and simultaneously enhance the chiral properties of the molecules, they can be coupled to metallic nanoparticles.9−11 It is also possible to shift optical activity to visible and infrared frequencies via the arrangement of achiral nanocrystals in pyramids, helices, and other chiral forms.12−14 Yet another way to produce optical activity is to nanostructure © XXXX American Chemical Society
Received: November 14, 2014 Revised: January 22, 2015
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Figure 1. Band spectrum of electrons in quantum nanowires (a) without and (b) with a right-hand screw dislocation b = 2a. It was assumed that R = 20a and the energy is in the units of ℏ2/(2ma2). (c) Schematic of electron transitions illustrating the absence of circular dichroism upon intraband dipole absorption of circularly polarized light by the dislocation−distorted nanowire.
dislocation-induced elastic deformation of the nanocrystal, the pure geometrical travel of electrons in the topologically distorted lattice, and the nanocrystal’s confinement, respectively. This equation neglects the Eshelby twist, which is the nanocrystal’s torque around the screw dislocation’s axis due to the stress and strain induced by the dislocation.40 If the screw dislocation is oriented along the z-axis, then its potentials in cylindrical coordinates are U1 = −2b2/(2πar)2 and U2 = b/(πr2) ∂φ∂z, where a is the undistorted-lattice constant and b is the z component of the Burgers vector. The symmetry and height of the confining potential are determined by the shape, size, and material of the nanocrystal, as well as by the host material. As we mentioned earlier, optical activity can originate from the dissymmetry of V caused by the nanocrystal shape but, as long as such a dissymmetry is absent, the exact form of V is insignificant for the emergence of the nanocrystal’s chirality. With this in mind, we take V as an infinitely large potential barrier at r = R to model a cylindrical nanowire of an impenetrable surface. We then find the wave function obeying eq 1 inside the nanowire to be given by41
for a deeper understanding of the optical activity of various kinds of dislocation−distorted quantum nanostructures, which are nowadays routinely fabricated with various techniques.33,34 It assumes that the nanocrystal chirality comes from the dissymmetry of the dislocation-induced potential affecting the confined electrons. We go beyond the Rosenfeld approximation and take into complete account the retardation of the electric field over the volume occupied by the nanocrystal. We show that the presence of a screw dislocation in a cylindrical nanowire inevitably leads to the circular dichroism of its absorption bands. This result suggests dislocation-enabled semiconductor nanocrystals as versatile building blocks for the creation of next-generation optically active materials with controllable properties. Screw Dislocations in Semiconductor Nanocrystals. The topological distortion of an ideal nanocrystal lattice by a screw dislocation breaks the lattice symmetry and alters the electronic subsystem of the nanocrystal. Being mostly pronounced on the length scale much larger than the atomic spacing, the distortion predominantly affects the long-wavelength quantum states of the nanocrystal. These states are described by the products of a dislocation-modified envelope wave function, ψ(r), and a Bloch amplitude, u(r). The modification of envelopes gives rise to the optical activity upon both interband35,36 and intraband transitions whereas the modification of amplitudes is usually neglected. To address the effect of screw dislocation on the nanocrystal chirality, we focus on the intraband transitions of electrons in semiconductor nanocrystals with a simple-cubic lattice. Using the two-band model of semiconductor and describing the dislocation within the frame of the differential-geometric approach based on the continuum theory of defects in solids,37 we require the longwavelength quantum states of the confined electrons to obey the Schrödinger equation38,39 (Δ + U1 + U2 + V )ψ = −εψ
ψnlk =
1 Jλ (ξnλr /R ) i(kz + lφ) e 2 πR Jλ + 1(ξnλ)
(2)
where ξnλ is the nth (n = 1, 2, 3,...) zero of the Bessel function Jλ(u), l = 0, ±1, ±2,... is the angular momentum, λ = [l2 + blk/π + 2b2/(2πa)2]1/2, and k is the wavenumber. The product blk in the Bessel function order tells us that the screw dislocation couples the radial, angular, and axial motions of electrons while discriminating between the quantum states of opposite handednesses. These features are reflected in the nanowire’s energy spectrum, which comprises the size2 quantized subbands εnl(k) = k2 + R−2ξnλ(l,k) . Ten subbands of an undistorted nanowire and a nanowire with a right-hand screw dislocation b = 2a are plotted in Figure 1a,b. We see that the dislocation blue shifts all the subbands with respect to their positions in the dislocation-free nanowire and removes the double degeneracy of the subbands with nonzero angular momentum. Note that the zero-momentum subbands are
(1)
where Δ is the Laplacian, ε = 2mE/ℏ2 is the reduced electron energy, and the potentials U1, U2, and V represent the B
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Figure 2. Theoretically simulated CD spectra of ZnS nanowire with a right-hand screw dislocation b = a and R = 20a for three kinds of intraband transitions. Insets show variations of angular momentum upon the transitions of each kind.
parabolas with minima at the Brillouin zone center, whereas the pairs of the dislocation-split subbands are nonparabolic and have equal minima located symmetrically with respect to the origin. We unambiguously demonstrate below that the dislocation-induced degeneracy removal of the electronic subsystem is the cause of circular dichroism in topologically distorted nanowires. Circular Dichroism Theory. Circular dichroism (CD) of a semiconductor nanowire is the difference in its absorption efficiencies of the left-hand-polarized and right-hand-polarized light.5 The dipole absorption efficiency is often calculated by neglecting the finiteness of the photon momentum and the associated inhomogeneity of the electric field over the nanowire volume. The conservation of electron’s wavenumber k, following from this approximation, essentially kills the CD of nanowires with screw dislocations. Indeed, if a circularly polarized photon excites an electron from subband (n, l) to subband (n′, l′), then a photon of opposite handedness and same energy excites with equal probability an electron of opposite wavenumber from subband (n,−l) to subband (n′, −l′), resulting in essentially zero CD. This statement is illustrated by Figure 1c, showing the two kinds of the dipoleallowed transitions between the subbands of zero and unity momenta. Note that CD is also absent in the less crude approximation in which the absorption efficiency of a monodisperse ensemble of randomly oriented nanowires is calculated by taking into account the electric quadrupole and magnetic dipole contributions. The measure of optical activity of such an ensemble upon a transition |nlk⟩ → |n′l′k′⟩ is the rotary transition strength defined as5,6
Im(⟨nlk|r|n′l′k′⟩·⟨n′l′k′|r × p|nlk⟩)
(3)
When the matrix elements in this scalar product are evaluated using the wave functions given in eq 2, the rotary strength vanishes due to different selection rules of l for the coordinate and momentum operators. The proper way to calculate CD of a nanowire whose length is not small compared to the wavelength of light is to consider the exact spatial variation of the electric field over the nanowire’s volume.8 To do so, we assume that excitation light propagates along the nanowire and describe its interaction with the confined electrons by the Hamiltonian42 ⎛∂ i ∂ ⎞ H ± = H0ei(qz ∓ φ)⎜ ∓ ⎟ r ∂φ ⎠ ⎝ ∂r
(4)
in which H0 is the electron−photon interaction strength, q is the z component of the photon wave vector, and the upper and lower signs correspond to the right-hand and left-hand polarized light, respectively. It is convenient to investigate CD upon intraband transitions using a pump−probe scheme. We shall therefore assume an experiment in which a linearly polarized optical pump generates electrons in a subband of zero momentum or in a pair of subbands (n, ±l), after which the electrons are excited to the higher-energy subbands by a weak circularly polarized probe. The subbands of opposite momenta are considered equally populated with total electron density N determined by the pump power. The distribution of electrons over wave numbers in subband (n, l) at temperature T is given by the Fermi function C
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Figure 3. Theoretically simulated CD spectra upon intraband transitions (1, ±2) → (1, ±3) in ZnS nanowires with (a) right-hand (solid curves) and left-hand (dashed curves) screw dislocations, R = 20 a and Na = 0.01 at T = 10 K; (b) b = a and Na = 0.01 at T = 10 K; (c) R = 20a, b = a and Na = 0.01; (d) R = 20a and b = a at T = 10 K. Dotted lines in (a) and (b) are the absorption spectra of linearly polarized light. Inset in (d) shows the relevant electronic subbands and distributions f nl(k) for three concentrations of electrons. Solid red spectra are the same in all cases.
⎡ ⎛ ε (k ) − ε ⎤−1 ⎞ nl 0 fnl (k) = 2⎢exp⎜ − μ⎟ + 1⎥ ⎠ κT ⎣ ⎝ ⎦
momentum subbands to subbands with a smaller momentum magnitude, |l′| = |l| − 1. All spectra have the shape of a few deviation peaks from the zero level and characteristic widths ranging from 0.03 to 0.3 meV. We see that the CD upon transitions of the first kind, illustrated by the spectra in Figure 2a−c, grows smaller as the principal quantum number n of the final electron state increases. This is a reflection of the general fact that the overlap integral of the radial part of the envelope wave function and its derivative decreases with |n′ − n|. Note that the spectral features observed are separated by large intervals of an almost zero CD signal, for example, from 12 to 62 meV and from 63 to 144 meV for intraband transitions from subband (1, 0). Transitions of the second kind, represented by the spectra in Figure 2d−f, show CD signals that are about hundred times stronger than those resulting from transitions of the first kind, whereas the signals generated upon transitions of the third kind in Figure 2g−i are only ten times stronger. Such a large variation of the CD signal strength is a consequence of the constructive and destructive interferences of probability amplitudes describing transitions due to the radial and angular derivatives in the electron−photon Hamiltonian. The constructive interference occurs for transitions of the second kind such that left-hand polarized light increases the positive angular momentum and right-hand polarized light decreases the negative momentum, whereas the destructive interference takes place for transitions of the third kind upon which the negative momentum is increased by left-hand polarized light and the positive momentum is reduced by right-hand polarized light (see insets in Figure 2). A notable difference between the CD upon transitions of the first group and those coming from transitions of the second and third groups is the opposite trends in their strength variations
(5)
where ε0 is the minimum energy of subband (n, l), μ is the effective chemical potential, κ = 2mkB/ℏ2, and kB is the Boltzmann constant. By neglecting depletion of the pumpexcited subbands and assuming that the rest of the subbands remain vacant due to the low probe power and fast intraband relaxation, we can write the intraband absorption rate of circularly polarized light using the Fermi golden rule43 as W± ∝
∫ dk ∫ dk′ ∑ ∑ fnl (k)
n′l′k′ H ± nlk 2 δ(ω − ωnlk ; n ′ k ′ l ′)
±l n ′ l ′
(6)
where ωnlk;n′k′l′ = ℏ[εn′l′(k′) − εnl(k)]/2m. This expression involves summations and integration over all the final states |n′l′k′⟩ shifted from the initial state |nlk⟩ by the photon energy ℏω, summation over the pump-excited subbands with opposite angular momenta and averaging over the energy distribution of the excited electrons. As expected, evaluation of this expression with the envelope wave functions of electrons in a dislocationfree nanowire (with b = 0) gives W+ = W−. Details of the analytical calculation of CD, ΔW = W− − W+, in the presence of a screw dislocation can be found in Supporting Information. Results and Discussion. Figure 2 shows CD spectra upon different intraband transitions in ZnS nanowire with a righthand screw dislocation b = a, R = 20a and Na = 0.01 at T = 10 K (for parameters of ZnS see Supporting Information). It is seen that all transitions can be classified into three groups according to the strength of their CD: (i) from a zeromomentum subband to subbands with l = ±1; (ii) from a pair of nonzero-momentum subbands to subbands with a larger momentum magnitude, |l′| = |l| + 1; and (iii) from nonzeroD
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is of the order of unity near the intraband resonance and equal unity out of the resonance. Such large g-factors are indicative of an extremely pronounced CD of the nanowires, much more pronounced than the molecular CD, which is significantly weaker than absorption. This is a consequence of infinite nanowire length and relatively simple treatment of intraband transitions. A measurement of CD and g-factor of real semiconducting nanowires with screw dislocations is still pending. Concluding Remarks. We have demonstrated that optical chirality is induced by a screw dislocation in a cylindrical nanowire of length much larger than the diameter. This conclusion still stands for true nanocrystals, with all spatial dimensions in the nanometer scale and fully discrete spectra. Although the result that CD originates from screw dislocations is general in nature, it is the easiest to demonstrate this fact for the special case of an impenetrable circular nanorod. This can be done by analytically solving eq 1 without the kinetic part of (0) the dislocation potential, (Δ + U1 + V)ψ(0) j = −εjψj and then treating the effect of U2 perturbatively. In the first order of perturbation theory, the wave functions of electrons inside the nanorod are given by
with probe frequency. The CD signal is seen to weaken with the frequency of intraband transitions 0 → ± 1, and grow stronger with the frequency of the other transitions. Also noteworthy is that the right-hand screw dislocation results in negative CD peaks upon transitions of the first and second kinds, and in positive peaks upon transitions of the third kind between the states of nonzero momenta. Hence, the maximum optical chirality occurs upon transitions between the electronic subbands with minimal |n′ − n| and |l′| > |l|. The nature of the modification of the CD spectrum is illustrated by Figure 3. In panel a, we see that an increase of the dislocation strength (Burgers vector magnitude) red shifts, broadens, and reduces the CD spectrum whereas a change of the dislocation handedness inverts the spectrum. Because neither dislocation strength nor handedness change the nanowire volume, the integral absorption is preserved. Panel b of the figure shows that the CD spectrum narrows with the nanowire’s radius and red shifts according to the well-known 1/ R2 law peculiar to the size-quantized energy spectra of semiconductor nanowires.41 In each case, the strongest CD dip (peak) is nearly at the position of the absorption resonance of linearly polarized light shown by the dotted curve. Figure 3c,d shows how the CD spectrum changes with the nanowire temperature and electron density, which are responsible for the energy distribution of electrons in the initial subband. We see from panel c that the spectrum broadens with temperature due to the rise in the number of electrons with large wave numbers. The room-temperature spectrum corresponds to the situation where the thermal energy (about 26 meV) exceeds the energy separation of the subbands and, thus, thermal excitation of the subbands needs to be taken into account. Considering such excitation would reduce the intensity and smear the peaks on the low-energy side of the CD spectrum (slightly above and below 26 meV) while almost not affecting its high-energy side. Panel d shows that higher densities of electrons generated by the pump yield stronger CD signal while only slightly affecting its spectral shape and position (note the quantitative difference between the three spectra, two of which are magnified by factors 2 and 100). The qualitative change of the spectral shape becomes noticeable only when the energy distribution turns step-like, as shown in the inset for Na = 1. According to Figure 3a,b, the CD of our nanowires is comparable in strength to the intraband absorption rate. The ratio of the two, known as the CD g-factor5 gCD = (W− − W+)/ (W− + W+), is plotted in Figure 4. One can see that the g-factor
ψj =
ψj(0)
+
∑ i≠j
⟨ψi(0) U2 ψj(0)⟩ εj − εi
(7)
where the subscripts i and j denote the sets of quantum numbers of electrons. By expanding the exponential in the electron photon interaction Hamiltonian as eiqz ≈ 1 + iqz and using the perturbed wave functions in eq 5, it is possible to calculate the strength of CD for different intraband transitions inside the nanorod. The dislocation-induced optical chirality of semiconductor nanorods will be addressed in detail in our future publications elsewhere. The CD spectrum of interband transitions will be the subject of detailed theoretical study elsewhere. Such a study may require more complex models of semiconductor bands, considering the anisotropic nature of the effective mass of holes, as well as taking into account the modification of the Bloch amplitudes. Our study suggests that most of the previously reported semiconducting nanowires and their complexes18,44−46 could have exhibited pronounced optical activity. This activity was not observable with the standard methods of the CD spectroscopy, as the nanowires of opposite handednesses form racemic mixtures of equal amounts of enantiomers, due to equal probabilities of formation of the left-handed and righthanded chiral features. Our study is, thus, of particular importance for the future development of semiconducting nanocrystals for photonics and nanobiotechnological applications. We believe that the CD-active semiconducting nanowires can potentially serve as new emitters of circularly polarized light, chemo-biosensors, and nanoprobes for molecular recognition of various chiral molecules, including biopharmaceutical products and common drugs. Our findings will also prove useful in a highly topical area of nanotoxicology, as most of biomolecules and biological species are chiral, interacting differently with right-handed and left-handed nanocrystals. These findings should thus be taken into account in the future research on the toxicity of nanostructures.
Figure 4. Theoretically calculated CD g-factor upon transitions (1, ±2) → (1, ±3) in ZnS nanowires for the same material parameters as in Figure 3a. E
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ASSOCIATED CONTENT
S Supporting Information *
Quantum states of semiconductor nanowire with a screw dislocation; electron−photon interaction; intraband transition rates; circular dichroism strength; and material parameters. This material is available free of charge via the Internet at http:// pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank the Science Foundation Ireland for Grant SFI 12/IA/1300, the Ministry of Education and Science of the Russian Federation for Grants 3.17.2014/K and 14.B25.31.0002, and the scholarship of the President of the Russian Federation for young scientists and graduate students (2013−2015). The work of I.D.R. is funded by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055. I.D.R. and A.S.B. also gratefully acknowledge the Monash Researcher Accelerator Program and the Dynasty Foundation Support Program for Physicists.
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