For a semiconductor nanocrystal to be chiral and optically active, it must lack a plane or center of symmetry. The minimal number of identical impurities required to break the mirror symmetry of a nanocrystal is the largest for a nanosphere. It is easy to see that regardless of their positions, neither one nor two impurities are sufficient to make a nanosphere chiral, because there is always a rotation axis of infinite order associated with one impurity and a mirror plane associated with two. A nanosphere may acquire chirality, provided it is doped with a minimum of three impurity ions, which form a plane that does not pass through the nanosphere’s center. Similarly, a minimum of two impurities are needed to break the mirror symmetry of a nanocylinder, and one to bring chirality to a nanocuboid.

The optical activity of doped semiconductor nanocrystals results from the perturbation of their electronic subsystems by the electric field of the impurities. Figure 1 illustrates this perturbation by the example of two electronic states and one impurity ion inside a nanocuboid. One can see that by breaking the mirror symmetry of the nanocuboid, the impurity makes the electronicwave functions essentially chiral. Thus the induced chirality shows up in the nanocrystal’s CD spectrum
$$\Delta {\epsilon}_{a}(\omega )=\frac{8\pi N\omega}{\hslash c}{\displaystyle \sum _{n}{\rho}_{ba}}(\omega ){R}_{ba},$$(1)

Figure 1 Wave functions of electronic states (112) and (224) inside a 3.4 × 3.8 × 4.2 nm^{3} nanocuboid (a) without and (b) with an impurity ion located at point *x* = 0.68 nm, *y* = 0.76 nm, and *z* = 0.84 nm (shown by dot). For visualization purpose, an ion of charge +10*e* is used. The wave functions are normalized by the volume of the nanocuboid.

where *N* is the concentration of the nanocrystals, *c* is the speed of light in vacuum, *ρ*_{ab}(*w*) is the spectral lineshape of transition *a* → *b*, and the rotatory transition strength is defined as [23]
$${R}_{ba}=\text{Im}({p}_{ab}\cdot {m}_{ba}),$$(2)

where $p=-er,m\text{=}-i\sqrt{\epsilon}e\hslash /(2{m}^{*}c)r\times \nabla ,\epsilon $ is the high-frequency permittivity, and *m** and *-e* are the effective mass and charge of an electron, respectively Note that the so defined rotatory strength does not depend on the polarization of the excitation light and describes ensembles with random nanocrystal orientations such as colloidal quantum dots.

In order to describe the quantum states of electrons confined by a doped semiconductor nanocrystal, we apply a one-band model of bulk semiconductor and ignore the exciton effects by assuming that the characteristic size of the nanocrystal, *l*, is much smaller than the exciton Bohr radius ${r}_{\text{B}}^{\text{ex}}={\epsilon}_{0}{\hslash}^{2}/({m}_{\text{ex}}{e}^{2})$, where £_{0} is the static permittivity and *m*_{ex} is the effective mass of an exciton. We also assume that the nanocrystal is made of a cubic semiconductor with an isotropic material response, characterized by scalar permittivities, and treat the impurities as point elementary charges. If the impurities are located at points **R**_{j}, then their impact on the confined electrons is described by an electric potential
$$V(\text{r})=-{\displaystyle \sum _{j}\frac{{e}^{2}}{{\epsilon}_{0}\left|{\mathrm{r-R}}_{j}\right|\text{'}}}$$(3)

which can be considered as a small perturbation of the nanocrystal. The envelope wave functions and energy states of the confined electrons are characterized by a set of three quantum numbers **n** and obey the Schrodinger equation
$${E}_{\mathbf{n}}{\psi}_{\mathbf{n}}=(\mathcal{H}+V){\psi}_{\mathbf{n}}$$(4)

in which the Hamiltonian of the impurity-free nanocrystal, $\mathcal{H}$, is achiral.

We next take the solution of Eq. (4) in the form of the standard perturbation expansion *E*_{n} = ε_{n} + V_{nn} and *Ψ*_{n} = *ψ*_{n} + *S*_{nn'} *ψ*_{n'}, where *ψ*_{n} and *ε*_{n} are the wave functions and energy spectrum of the unperturbed Hamiltonian $\mathcal{H},{V}_{{n}^{\prime}n}={\displaystyle \int {\psi}_{{n}^{\prime}}^{*}V}{\psi}_{n}\text{d}r$ and we have introduced a summation operator ${S}_{{n}^{\prime}n}={\displaystyle {\sum}_{n}^{\text{'}}{V}_{{}_{{n}^{\prime}n}}}/{\epsilon}_{n}-{\epsilon}_{{n}^{\prime}}$. In writing the expansion in the form shown, we have assumed that the states of the unperturbed Hamiltonian are non-degenerate. It can be shown that this expansion is accurate for a nanocrystal with *Z* impurity ions, provided that the size of the nanocrystal *l* ≪ *r*_{B}(3π)^{2}/(2*Z*), where *r*_{B} = ε_{0}*ℏ*^{2}/(*m***e*^{2}) is the conductivity electron Bohr radius

Let us focus on studying the optical activity of a semiconductor nanocuboid *l*_{x} × *l*_{y} × *l*_{z}, whose optical activity can be achieved with a single impurity ion. By assuming that the nanocrystal is located symmetrically with respect to the three Cartesian planes and its surface is impenetrable for the confined electrons, we take ${\psi}_{n}(\text{r}){\prod}_{v}\sqrt{2/{l}_{v}}{H}_{v}({k}_{v}v)$ and ${\epsilon}_{n}={\hslash}^{2}({k}_{x}^{2}+{k}_{y}^{2}+{k}_{2}^{2})/(2{m}^{*})$, where **n** = (*n*_{x}, *n*_{y}, *n*_{z}), *H*_{v}(*ξ*) = cos *ξ* for odd *n*_{v}, *H*_{v}(*ξ*) = sin *ξ* for even *n*_{v}, and *k*_{v} = *k*_{nv} = *π**n*_{v}/*l*_{v}. With these wave functions, the *i*-th components of matrix elements **p**_{nn}" and **m**_{n"n} are found to be given by
$$\begin{array}{l}\int {\psi}_{\mathbf{n}}{p}_{i}{\psi}_{{\mathbf{n}}^{\u2033}}\mathbf{d}\mathbf{r}=-e{l}_{i}({B}_{{n}_{i}{{n}^{\u2033}}_{i}}{\delta}_{{n}_{i}{{n}^{\u2033}}_{i}}{\delta}_{{n}_{k}{{n}^{\u2033}}_{k}}\\ +{s}_{\mathbf{n}{\mathbf{n}}^{\prime}}{B}_{{n}_{i}{{n}^{\u2033}}_{i}}{\delta}_{{n}_{i}{{n}^{\u2033}}_{i}}{\delta}_{{n}_{k}{{n}^{\u2033}}_{k}}+{s}_{{\mathbf{n}}^{\u2033}{\mathbf{n}}^{\prime}}{B}_{{n}_{i}{{n}^{\u2033}}_{i}}{\delta}_{{n}_{i}{{n}^{\u2033}}_{i}}{\delta}_{{n}_{k}{{n}^{\u2033}}_{k}}\end{array}$$(5a)

and
$$\begin{array}{l}\int {\psi}_{{\mathbf{n}}^{\u2033}}{m}_{i}{\psi}_{\mathbf{n}}\mathbf{d}\mathbf{r}=-i\frac{\sqrt{\epsilon}e\hslash}{2{m}^{\ast}c}{\epsilon}_{ijk}({\delta}_{{n}_{i}{{n}^{\u2033}}_{i}}{G}_{{n}_{k}{{n}^{\u2033}}_{k}}^{{n}_{k}{{n}^{\u2033}}_{k}}\\ +{S}_{\mathbf{n}{\mathbf{n}}^{\prime}}{\delta}_{{n}_{i}{{n}^{\u2033}}_{i}}{G}_{{n}_{k}{{n}^{\u2033}}_{k}}^{{n}_{k}{{n}^{\u2033}}_{k}}+{S}_{{\mathbf{n}}^{\u2033}{\mathbf{n}}^{\prime}}{G}_{{n}_{k}{{n}^{\prime}}_{k}}^{{n}_{k}{{n}^{\prime}}_{k}}),\end{array}$$(5b)

where *i* ≠ *j* ≠ *k*, *δ*_{ij} and *ε*_{ijk} are the Kronecker's delta and the Levi-Civita symbol, and
$${B}_{{n}_{i}{{n}^{\prime}}_{i}}=\frac{8{n}_{i}{{n}^{\prime}}_{i}}{{\pi}^{2}{({n}_{1}^{2}-{{n}^{\prime}}^{2}{}_{i})}^{2}}\text{Re\hspace{0.17em}}{i}^{{n}_{i}+{{n}^{\prime}}_{i}+1},$$(6)
$${G}_{{n}_{k}{{n}^{\u2033}}_{k}}^{{n}_{k}{{n}^{\u2033}}_{k}}=\frac{{l}_{j}{l}_{k}}{2}{B}_{{n}_{j}{{n}^{\prime}}_{j}}{B}_{{n}_{k}{{n}^{\prime}}_{k}}\left({k}_{{n}_{j}}^{2}-{k}_{{{n}^{\prime}}_{j}}^{2}-{k}_{{n}_{k}}^{2}+{k}_{{{n}^{\prime}}_{k}}^{2}\right),$$(7)

The selection rules for intraband transitions are determined by Kronecker deltas, coefficients ${\text{B}}_{{n}_{1}{{n}^{\prime}}_{1}}$ (which are nonzero only when *n*_{i} and ${{n}^{\prime}}_{1}$
are of different parities), and coefficients ${G}_{{n}_{j}{{n}^{\u2033}}_{k}}^{{n}_{k}{{n}^{\u2033}}_{k}}$. It is readily seen that *R*_{ba} = 0 if the im-purities are absent, because of the different selection rules for the electric and magnetic moment operators.

The obtained result leads to a simple size dependence of the rotatory strength. Suppose that the lengths of the nanocrystal edges and the coordinates of the impurities are increased by a factor of *f*. Then ε_{n} ∝ *f*^{-2}, *V* ∝ *f*^{-1} and *S*_{nn'} ∝ *f*. As the matrix element of the magnetic dipole moment does not change with *f* while the electric dipole moment scales as ∝ *f*, Eqs. (2) and (5) give *R*_{ba} ∝ *f*^{2}. One can see that the size dependence of the rotatory strength comes from three sources: scaling of the confinement energies, scaling of the impurity potential, and scaling of the electric dipole moment. As the nanocrystal grows bigger, the Coulomb interaction between electrons and holes more and more reduces the confinement energies, resulting in their weaker size dependence *ε*_{n} ∝ *f*^{-q}, where 1 < *q* < 2. Hence, the rotatory strength in the weak confinement regime is larger than that in the strong confinement regime but has a weaker size dependence, *R*_{ba} ∝ *f*^{q}.

Equations (2) and (5) show that there are six types of dipole-allowed transitions, two per each Cartesian axis. In particular, the confined motion along the *x*-axis leads to electric dipole transitions, which change the parity of state ψ *ψ*_{x} and preserve states *ψ*_{y} and *ψ*_{z}, and to magnetic dipole transitions, which preserve state *ψ*_{x} and change the parities of states *ψ*_{y} and *ψ*_{z}. The electric dipole mo-ment is much larger than the magnetic dipole moment for the transitions of the first kind, |**p**_{ab} | ≫ |**m**_{ba}|, and the opposite relation holds for the transitions of the second kind, |**m**_{ha}| ≫ |**P**_{ab}|.

The rotatory strength of transition $({n}_{x},{n}_{y},{n}_{z})\to ({{n}^{\u2033}}_{x},{{n}^{\u2033}}_{y},{{n}^{\u2033}}_{z})$ induced by the *x* components of the electric and magnetic dipole moments is determined by the expressions
$${B}_{{n}_{x}{{n}^{\u2033}}_{x}}\left({S}_{{n}_{x}{n}_{y}{n}_{z};{{n}^{\u2033}}_{x}{{n}^{\prime}}_{y}{{n}^{\prime}}_{z}}-{S}_{{{n}^{\u2033}}_{x}{n}_{y}{n}_{z};{n}_{x}{{n}^{\prime}}_{y}{{n}^{\prime}}_{z}}\right){G}_{{{n}^{\prime}}_{y}{n}_{y}}^{{{n}^{\prime}}_{z}{n}_{z}},$$(8a)
$${G}_{{n}_{y}{{n}^{\u2033}}_{y}}^{{n}_{z}{{n}^{\u2033}}_{z}}\left({S}_{{n}_{x}{n}_{y}{n}_{z};{{n}^{\prime}}_{x}{{n}^{\u2033}}_{y}{{n}^{\u2033}}_{z}}-{S}_{{n}_{x}{{n}^{\u2033}}_{y}{{n}^{\u2033}}_{z};{{n}^{\prime}}_{x}{n}_{y}{n}_{z}}\right){B}_{{n}_{x}{{n}^{\prime}}_{x}}.$$(8b)

The rotatory strength is seen to be contributed only by the sums involving the matrix elements of the perturbation-induced transitions between the states with different parities of the respective quantum numbers. For a nanocrys-tal doped with one impurity ion, these matrix elements are antisymmetric with respect to each ion's coordinate, making rotatory strength vanish in case the ion is located in one of the three Cartesian planes. This conclusion also holds for other components of the dipole moments, the rotatory strengths of which can be obtained from Eq. (8) by a cyclic change of subscripts *x*, *y*, and *z*. As the reflection of the impurity in any of the coordinate planes changes the sign of the rotatory strength, knowing *R*_{ba} for all impurity locations in the first octant of the nanocrystal allows us to find *R*_{ba} in the entire nanocrystal.

## 3 Results and discussion

Engineering optical activity of doped nanocrystals requires knowing the optimal positions ofimpurityions that maximize the activity. The optimal positions critically depend on the transition upon which the activity is observed, and can be estimated by analyzing the dominant terms in Eq. (8). These terms correspond to the perturbation-induced transitions between the unperturbed initial state *ψ*_{x, ny, nz} and the unperturbed final state ${\psi}_{{{n}^{\u2033}}_{x},{{n}^{\u2033}}_{y},{{n}^{\u2033}}_{z}}$ and the closest to them unperturbed states of the nanocrystal. To illustrate this point, consider six transitions from the ground state to states *Ψ*_{211}, *Ψ*_{121}, *Ψ*_{112}, *Ψ*_{122}, *Ψ*_{212}, and *Ψ*_{221}. For all of them, *ψ*_{222} is the closest state coupled by perturbation *V* to state *ψ*_{111}. The perturbation-induced transitions between these states give the first dominant contribution to the six rotatory strengths. If the three nanocrystal edges do not differ too much, then the second major contribution to *R*_{211;111}, *R*_{121;111}, ⋯ comes from transitions ${\psi}_{211}\rightleftarrows {\psi}_{122},{\psi}_{121}\rightleftarrows {\psi}_{212},\cdots \cdot $ Thema-trix elements of these transitions are all alike, *V*_{111;222} = *V*_{211;122} = ..., resulting in the same dependence of the six dominant terms on the impurity position. The locations and number of absolute maxima of rotatory strengths *R*_{211;111}, *R*_{121;111}, ... are therefore, determined by a single matrix element *V*_{111;222}. As the wave function product ${\prod}_{v}\mathrm{cos}(\pi v/{l}_{v})\mathrm{sin}(2\pi v/{l}_{v})$ has one maximum in the first octant, there is also a single absolute maximum of the six rotatory strengths.

The isosurfaces of *R*_{ba} shown in Fig. 2 illustrate the dependence of rotatory strength on the impurity position inside the first octants of nanorod, quantum dot, and nanoplatelet of approximately equal volumes. In agreement with the above discussion, the rotatory strength upon the electric dipole transition *Ψ*_{111} → *Ψ*_{121} peaks at one point, clearly seen in panels (a)-(c). The optimal impurity positions inside the nanorod, quantum dot, and nanoplatelet are (0.50, 0.44, 0.40), (0.46, 0.44, 0.42), and (0.50, 0.42, 0.40), respectively. According to panel (d), rotatory strength *R*_{141;111} has two maximums located at points (0.46, 0.22, 0.36) and (0.46, 0.72, 0.36) along the *y*-axis; they come from two maximums of the wave function product cos(*πu*_{y}|2)sin(2*πu*_{y}). Similarly, there are two and three maximums of *R*_{124;111} and *R*_{116;111} along the *z*-direction, as can be seen from panels (e) and (f).

Figure 2 Isosurfaces of rotatory strength *R*_{ba} (in the units of 10^{-39} erg × cm^{3}) in the first octant of 2 × 3 × 9 nm^{3} nanorod, 3.4 × 3.8 × 4.2 nm^{3} quantum dot, and 1.5 × 5 × 6 nm^{3} nanoplatelet. Transitions occur from the ground state (111) to state (121) in panels (a)-(c) and to states (141), (124), and (116) in panels (d), (e), and (f), respectively. All the plots are in normalized coordinates *u*_{v} = 2*v*|*l*_{v} (*v* = *x*, *y*, *z*). The modeling material is InP and the simulation parameters are: ε_{0} = 12.35, ε = 9.61, *m** = 0.08 *m*_{0} (*m*_{0} is the free-electron mass), and *r*_{B} = 8.12 nm.

One can see that the optical activity of semiconductor nanocrystals doped with one ion of charge *+e* can be 10 times larger than the rotatory strengths of chiral molecules, which typically range from 10^{-39} to 10^{-38} erg × cm^{3} [24-26]. Using several dopants of bigger charges, optimally positioned in the third, sixths, and eighths octants, one can increase the rotatory strengths of optical transitions by another factor of 10. It should also be noted that the rotatory strength is maximal for the lowest-energy intraband transitions and steeply decreases with transition energy becouse of the decrease in the probabilities of the electric and magnetic dipole transitions, governed by the coefficients ${B}_{{n}_{i}{{n}^{\prime}}_{i}}$ and ${G}_{{n}_{j}{{n}^{\prime}}_{j}}^{{n}_{k}{{n}^{\prime}}_{k}}$

Figure 3 CD spectra of (a) 2 × 3 × 9 nm^{3} nanorod, (b) 3.4 × 3.8 × 4.2 nm^{3} quantum dot, and (c) 1.5 × 5 × 6 nm^{3} nanoplatelet with different positions of an impurity shown by the color dots in the insets. The insets represent cross sections of the nanocrystals at *x* = *l*_{x}/4. All the spectral lines are approximated by Lorentzians with a FWHM of 0.1 eV [27]. The material parameters are the same as in Fig. 2.

Despite the fact that the rates of electric dipole and magnetic dipole transitions differ by a factor of 10,000 [28], the respective rotatory strengths are of the same order of magnitude. The large difference in the transition rates shows up in dissymmetry factor *g*, which is the doubled ratio of CD to the sum of absorption intensities for left- and right-handed circularly polarized light. This feature is illustrated in . The large absorption upon the electric dipole transition is seen to result in *g*-factors in the range from 10^{-4} to 10^{-3}, which is typical for small chiral molecules [29]. As the magnetic dipole absorption is about 10^{4} weaker than the electric dipole one, the *g*-factors of the magnetic dipole transitions can be of the order of unity, indicating a strong chiroptical response. Note that the g-factors in are not maximal, as they are calculated for the impurity position providing maximum to the rotatory strength.

Table 1 Rotatory strengths (in the units of erg × cm^{3}) and dissymmetry factors of (a) electric dipole transition (111) → (121) and (b) magnetic dipole transition (111) → (122) inside nanorod, quantum dot, and nanoplatelet with a single impurity located at optimal position **u**, which corresponds to the maximum of rotatory strength. All material parameters are the same as in Fig. 2.

The dependence of the CD spectra of doped nanocrys-tals on the impurity position is illustrated in Fig. 3. The impurity is assumed to be in plane *x* = *l*_{x}/4, where its positions are shown by the color dots in the insets. The optical CD spectrum of the nanorod in Fig. 3(a) has two strong peaks, corresponding to the transitions in , and a much weaker peak of transition *Ψ*_{111} → *Ψ*_{124}. As the impurity is moved from left to right, the weaker peak changes its sign, while nearly disappearing when the dominant peaks have maximum intensity. The CD spectrum of the quantum dot in Fig. 3(b) consists of six equally pro-nounced dipole transitions, whose rotatory strengths were shown to exhibit almost the same dependence on the impurity position. All of the peaks are the strongest where the impurity is located about the center of the first octant. The optical CD spectrum of the nanoplatelet shown in Fig. 3(c) includes four transitions, whose intensities depend on the impurity position as it is predicted by Fig. 2(d) and (e). There are also three low-energy peaks in the spectrum, which vary with position similar to the states of the quantum dot.

## 4 Conclusion

We have shown that one can control and maximize the optical activity of semiconductor nanocrystals by introducing impurity ions inside them. The proposed approach allows one to achieve rotatory strengths upon intraband transitions that are 100 times larger than the typical rotatory strengths of small chiral molecules. It also enables re-alization of an almost complete dissymmetry of magnetic dipole absorption, opening a new chapter in the studies of light-matter interaction on the nanoscale.

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