Giant Optical Activity of Quantum Dots, Rods, and Disks with Screw Dislocations

For centuries mankind has been modifying the optical properties of materials: first, by elaborating the geometry and composition of structures made of materials found in nature, later by structuring the existing materials at a scale smaller than the operating wavelength. Here we suggest an original approach to introduce optical activity in nanostructured materials, by theoretically demonstrating that conventional achiral semiconducting nanocrystals become optically active in the presence of screw dislocations, which can naturally develop during the nanocrystal growth. We show the new properties to emerge due to the dislocation-induced distortion of the crystal lattice and the associated alteration of the nanocrystal’s electronic subsystem, which essentially modifies its interaction with external optical fields. The g-factors of intraband transitions in our nanocrystals are found comparable with dissymmetry factors of chiral plasmonic complexes, and exceeding the typical g-factors of chiral molecules by a factor of 1000. Optically active semiconducting nanocrystals—with chiral properties controllable by the nanocrystal dimensions, morphology, composition and blending ratio—will greatly benefit chemistry, biology and medicine by advancing enantiomeric recognition, sensing and resolution of chiral molecules.

The solution to the Schrödinger equation without the kinetic term and with the confinement potential of the form is given by energies

First-order perturbation theory for the kinetic term
The kinetic term = /( ) 2 can be treated as a small perturbation of the nanocrystal's electronic subsystem. Since the Bessel function order ( ) is nonzero for all , the matrix elements of perturbation are all finite and given by if and ′ have different parities, and ′ = 0 if and ′ are of the same parity.
The first-order energies and wave functions of the nanocrystal electrons are of the form The energy correction is zero because

Applicability limits of the perturbation theory
Consider the smallness parameter Since is the only quantum number always changed by the perturbation, the denominator of can have either four or two terms, depending on whether the principal quantum number is changed or not: Let us first assume that the principal quantum number does not change. Then For ′ < and ′ < we have For ′ > and ′ > we have These conditions are satisfied automatically, as 3 2 ≫ 2| | for all real nanocrystals with screw dislocations.
For ′ > and ′ < , as well as for ′ < и ′ > , we have The denominator of this expression may be small only in case of accidental degeneracy of electronic states, i.e. if the energies of states with different quantum numbers , ′, and ′ coincide. In this paper, we assume that such an accidental degeneracy is absent.

Interaction of electrons with circularly polarized light
To study intraband transitions in our nanocrystal, we employ the representation of lightmatter interaction. Consider left-hand and right-hand circularly polarized plane waves of wave vector = √ / propagating along the axis. The interaction of electrons with such waves can be represented by operators The matrix elements of implicitly defined operators ± upon transitions occurring without a change of the parity of quantum number are given by whereas the matrix elements upon transitions with the parity change are: It can be shown that We thus finally obtain

Circular dichroism
Consider intraband absorption of light by electrons of fixed energy generated through interband transitions by a linearly polarized pump. The energy of electrons can correspond to a nondegenerate state of zero angular momentum or a pair of doubly degenerate states of nonzero momentum. The rest of electronic states are assumed to be unoccupied, and the probe is assumed to be weak to prevent depletion of the excited state(s). Using Fermi's golden rule, it can then be shown that the difference of intraband transition rates due to lefthand and right-hand polarized light is given by Here the summation extends over all the final states that are excited from a given initial state ( ) by the probe of frequency .
It is convenient to consider separately transitions occurring with and without a change in the parity of the quantum number . Using the selection rules to evaluate the summation over the final quantum number ′ , we obtain in the latter case where it has been taken into account that ′ ± = ′ ,− ∓ . Hence, intraband transitions preserving the quantum number are optically inactive.
If the parity of changes, there are three kinds of intraband transitions: (ii) ± → ± ± 1; (iii) ± → ± ∓ 1. For the transitions of the first kind, we obtain And, finally, for the transitions of the third kind, we find It is seen that Δ can be written in the general form as and where the upper sign corresponds to the transitions of the first and second kinds and the lower sign corresponds to transitions of the third kind.

Averaging over the spatial orientations of nanocrystals
To take into account the fact that nanocrystals are oriented randomly in real systems, we must average the obtained result over all possible nanocrystal orientations in space. To do this, we fix the orientation of a nanocrystal in space while assuming that = (| | = 1) and the vector potential of the probe is in the plane of orthogonal unit vectors and . By employing our freedom in the choice of and associated with the circular polarization of light, we place in the -plane and set the orientation of the reference frame { , , } with respect to the frame ( , , ) by a pair of angles. If is the angle between and the axis and is the angle between the projection of vector to the -plane and the axis (see Fig. 1