Man-made semiconductor nanocrystals, whose shapes are often approximated by spheres, cubes, or pyramids, are, in fact, more or less irregular nano-objects due to various kinds of surface and bulk defects. These defects occur both naturally and as a result of imperfections in fabrication techniques [1]. It has been recently shown that ZnS nanoplatelets, nanorods, and quantum dots become chiral and acquire optical activity in the presence of screw dislocations, which are naturally forming inside them during the fabrication process [2–4]. The emerging asymmetry of the nanocrystals’ interaction with left circularly polarized (LCP) light and right circularly polarized (RCP) light manifests itself in the circular dichroism (CD) spectrum, and is currently well understood [5,6]. In addition to directly inducing optical activity by distorting the crystal lattice, screw dislocations produce strain, relaxation of which leads to the Eshelby twist of the nanocrystal [7] and alters the envelope wave functions of the confined charge carriers. The irregularity of the nanocrystal’s surface contributes to its optical activity, and shows up as extra spectral lines in the CD spectrum.

Screw dislocations and the associated Eshelby twists are not the only defects, of course, that can make semiconductor nanocrystals chiral. Various other bulk and surface defects do break the mirror and inversion symmetries of nanocrystals, making them optically active. This implies that, strictly speaking, all semiconductor nanocrystals to certain extent are chiral [8]. Therefore, the practical challenge is not to fabricate a chiral nanocrystal, but rather to favor the formation of its specific enantiomer, so that the optical activity can be exhibited by a large nanocrystal ensemble. Theoretically, it is essential to be able to estimate the magnitude of chiroptical response associated with the natural irregularity of the nanocrystal surface and to explain all the features of the observed CD spectra. With this Letter, we provide a handy tool that enables one to solve this theoretical task and analytically estimate the strength of optical activity of a chiral semiconductor nanocrystal.

To analytically treat the optical activity of a chiral semiconductor nanocrystal, we employ the method of coordinate transformation [9]. This method relies on finding such a transformation of coordinates, $\mathbf{r}=\phi (\mathbf{R})$, that turns the nanocrystal of a chiral surface $S(\mathbf{R})$ into a nanocrystal of a simpler achiral surface $s(\mathbf{r})=S(\mathbf{R})$, where its electronic subsystem admits analytical treatment. While making the shape of the nanocrystal regular, the transformation also changes the nanocrystal’s Hamiltonian, $H(\mathbf{r})$, resulting in the additional potential $\delta V(\mathbf{r})=H[{\phi }^{-1}(\mathbf{r})]-H(\mathbf{r})$. If the achiral surface is close to the chiral one, $\delta V(\mathbf{r})$ can be treated using the stationary perturbation theory. By denoting as $|\mathbf{n}⟩$ and ${\epsilon }_{\mathbf{n}}$ the non-degenerate eigenfunctions and eigenvalues of $H(\mathbf{r})$, we can represent the corrected (to the first order of $\delta V$) eigenfunctions and eigenvalues of the transformed Hamiltonian $H(\mathbf{r})+\delta V(\mathbf{r})$ as

In this Letter we focus on semiconductor nanocrystals, which have chiral shapes close to nanocuboids, because the wave functions and energy spectrum of nanocuboids have the simplest form in the two-band approximation [11]. Indeed, if the surface of a semiconductor nanocuboid is impenetrable for its confined charge carriers, the solution to the Schrödinger equation $-{\nabla }_{xyz}^2|\mathbf{n}⟩={\epsilon }_{\mathbf{n}}|\mathbf{n}⟩$ is given by ${\epsilon }_{\mathbf{n}}=k_x^2+k_y^2+k_z^2$ and

Denote as $X$, $Y$, and $Z$ the ordinary Cartesian coordinates, in which our nanocrystal has an irregular chiral shape shown on the left of Fig. 1. Without loss of generality, we shall assume that the nanocrystal occupies the region of space defined by the inequalities

 

Fig. 1. Coordinate transformation $\mathbf{r}=\phi (\mathbf{R})$ turns a chiral semiconductor nanocrystal of irregular surface $S(\mathbf{R})$ into a nanocuboid of the same volume and surface $s(\mathbf{r})$.

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Mathematically, the nanocrystal exhibits optical activity only when functions $F$, $G$, and $H$ have certain parities with respect to their arguments. The allowable parities can be found by decomposing the rotatory strength $R_{\mathbf{mn}}=-i\rho (\mathbf{n}|\mathbf{r}|\mathbf{m})(\mathbf{m}|\mathbf{L}|\mathbf{n})$ upon intraband transition $|\mathbf{n})\to |\mathbf{m})$ into a sum of electric dipole (ED) and magnetic dipole (MD) contributions using Eq. (1) as follows:

Coefficients $B_{n_x{m}_x}$ and $G_{n_y{m}_y}^{n_z{m}_z}$ determine the selection rules of the ED and MD intraband transitions, and are nonzero only when $n_v$ and $m_v$ have opposite parities. Accounting for this in Eq. (8) shows that the two kinds of rotatory strengths do not vanish only if perturbation $V_{\mathbf{sm}}$ couples the states with different parities of all three quantum numbers. This will be the case given the form of $\delta V$ in Eq. (6), provided the functions in Eq. (7) have the following parities with respect to their arguments:

A nanocrystal is chiral when it cannot be superposed onto its own mirror image. Clearly, functions $f$, $g$, and $h$ in the form of a linear superposition of their two arguments (e.g., $f={\alpha }_{1}Y+{\alpha }_{2}Z$) describe an achiral object. Mathematically, such functions result in constant $F$, $G$, and $H$, and, thus, produce no optical activity. It also can be concluded that a chiral nanocrystal may exhibit optical activity without changing its energy spectrum with respect to the spectrum of nanocuboid, because only the nondiagonal matrix elements of $\delta V$ contribute to the rotatory strengths.

Let us now consider optical activity in the most general situation, in which the chirality of the nanocrystal is due to the twist of its all six facets. This situation is described by the shape functions $f=\alpha YZ$, $g=\beta ZX$, and $h=\gamma XY$, and by the perturbation potential $\delta V=(\beta +\gamma )X{\partial }_y{\partial }_z+(\alpha +\gamma )Y{\partial }_z{\partial }_x+(\alpha +\beta )Z{\partial }_x{\partial }_y$. The nanocrystals characterized by sets of parameters $\{\alpha ,\beta ,\gamma \}$ and $\{-\alpha ,-\beta ,-\gamma \}$ are the nonsuperposable mirror images of each other known as enantiomers. The transformation given in Eq. (5) is readily inverted to the first order of $\alpha$, $\beta$, and $\gamma$ with the result

 

Fig. 2. Three kinds of chiral nanocuboids whose optical activity can be described using the developed analytical approach: [(a) and (b)] two enantiomers with a pair of distorted facets $(\beta =\gamma =0)$; [(c) and (d)] nanocuboids with two pairs of distorted facets $(\gamma =0)$; and [(e)–(h)] nanocuboids with all facets distorted; $L_x=6\mathrm{nm}$, $L_y=8\mathrm{nm}$, $L_z=10\mathrm{nm}$, $\alpha =0.06{\mathrm{nm}}^{-1}$, $\beta =0.08{\mathrm{nm}}^{-1}$, and $\gamma =0.1{\mathrm{nm}}^{-1}$.

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The distortion of real semiconductor nanocrystals can be almost as pronounced as it is in Fig. 2. For example, a simple twist of a $3\mathrm{nm}\times 4\mathrm{nm}\times 5\mathrm{nm}$ ZnS nanocuboid $(\beta =\gamma =0)$, corresponding to the deviations of its corners by a lattice constant of 0.541 nm, results in $\alpha \approx 0.22{\mathrm{nm}}^{-1}$. Table 1 presents rotatory strengths, dissymmetry factors, and peak CD signals for six intraband transitions inside this nanocrystal with ${\epsilon }_{\infty }=8.5$ and $m=0.2m_0$, where $m_0$ is the free-electron mass. It was assumed that the peak CD signal is related to the rotatory strength as ${\mathrm{CD}}_{\mathbf{mn}}=8\pi N{\omega }_{\mathbf{mn}}{R}_{\mathbf{mn}}/(c\hslash {\mathrm{\Gamma }}_{\mathbf{mn}})$, where $N$ is the nanocrystal concentration, ${\omega }_{\mathbf{mn}}$ is the transition frequency, and $2{\mathrm{\Gamma }}_{\mathbf{mn}}$ is the full width at half-maximum (FWHM) of the spectral line. The data in the table corresponds to the typical values $N={10}^{16}{\mathrm{cm}}^{-3}$ and $\hslash {\mathrm{\Gamma }}_{\mathbf{mn}}=50\mathrm{meV}$ [12,13]. One can see that the optical activity of the nanocrystal is the strongest upon the ED transitions associated with the change in the confined motion along the smallest nanocrystal dimension $(L_x)$ and upon the MD transitions due to the motion in the perpendicular plane. Also noteworthy are the giant dissymmetry factors of the MD transitions, which are explained by the weaker MD absorption compared to the ED one.

Table 1. Rotatory Strengths (in the Units of ${10}^{-38}\mathrm{erg}\times {\mathsf{cm}}^3$), Dissymmetry Factors, and Peak CD Signals (in ${\mathsf{cm}}^{-1}$) for Three ED and Three MD Transitions Inside a Chiral ZnS Nanocrystal with $L_x=3\mathsf{nm}$, $L_y=4\mathsf{nm}$, $L_z=5\mathsf{nm}$, and $\alpha =0.22{\mathsf{nm}}^{-1}$

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Comparison of the optical activity of chiral quantum dots (QDs), nanorods (NRs), and nanoplatelets (NPLs) is also instructive. Such a comparison is provided by Table 2, which assumes the same material parameters as in Table 1. The optical activity is seen to be the strongest for the nanocrystals whose three dimensions differs the most. Despite the great difference between the dissymmetry factors of the ED and MD transitions, the respective rotatory strengths of the two are nearly the same for each kind of nanocrystal.

Table 2. Same as in Table 1 but for ED Transition $|111)\to |121)$ and MD Transition $|111)\to |212)$ inside Chiral ZnS Nanoplatelets, Nanorods, and Quantum Dots

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In conclusion, we have developed a general approach to studying optical activity of chiral semiconductor nanocrystals with irregularities of shapes due to the surface and bulk defects. Our approach allows calculation of the rotatory strengths, dissymmetry factors, and circular dichroism spectrum upon transitions inside a chiral nanocrystal, provided a linear transformation of coordinates transforms this nanocrystal into an achiral one with a simple regular surface. The developed approach was illustrated by the example of intraband transitions inside a nanocrystal whose shape is close to a nanocuboid. Our approach may prove useful for other nanocrystal shapes as well as for studying optical activity of chiral nanocrystals upon interband transitions.