Cooperative expression of atomic chirality in inorganic nanostructures

Cooperative chirality phenomena extensively exist in biomolecular and organic systems via intra- and inter-molecular interactions, but study of inorganic materials has been lacking. Here we report, experimentally and theoretically, cooperative chirality in colloidal cinnabar mercury sulfide nanocrystals that originates from chirality interplay between the crystallographic lattice and geometric morphology at different length scales. A two-step synthetic scheme is developed to allow control of critical parameters of these two types of handedness, resulting in different chiral interplays expressed as observables through materials engineering. Furthermore, we adopt an electromagnetic model with the finite element method to elucidate cooperative chirality in inorganic systems, showing excellent agreement with experimental results. Our study enables an emerging class of nanostructures with tailored cooperative chirality that is vital for fundamental understanding of nanoscale chirality as well as technology applications based on new chiroptical building blocks.


S19
Fitting Parameters for ε Fitting Parameters for ξ

Light-matter interaction involving chiral media
Our core-shell model (see Supplementary Fig. 1) incorporates a crystalline lattice with arbitrary geometric shape (that can be either chiral or achiral) as core particle, which is embedded in a spherical non-chiral dielectric environmental matrix shell. The dimension of non-chiral dielectric matrix shell is typically much larger than that of central nanostructure to account for the environmental media in a CD measurement. In general, the interaction between electromagnetic wave (such as light) and matter is describable in where k ! is incident wave vector, E !" 0 and H !" 0 are amplitudes of electric and magnetic fields, respectively. To perform a FEM simulation, a PML is also added in our model as the outermost shell to absorb launched waves at the boundary of dielectric medium, which simulates an infinite environment and helps truncate computational domains.
In our current model, we treat all our nanostructures as isotropic non-magnetic chiral media, in which the constitutive equations can be given by 3,4 : where ε BPE is the Bassiri-Papas-Engheta dielectric function and is related to ε BPE = ε − ξ 2 µ , and µ BPE = µ . In our model, we therefore adopt above constitutive relations with ε and µ independent of the chiral parameter ξ . This can allow us to evaluate chirality originating from crystallographic and geometric effects independently.
Furthermore, we consider only isotropic chiral lattice in our current work, that is, the ξ is isotropic. While in principle the should be anisotropic depending on symmetry group ξ S24 of a lattice, as a first order approximation, assumption of isotropy can simplify calculation as well as determination of parameters, and allow us to address underlying basic physics, but in the future, more accurate description of anisotropic ξ should further improve the agreement between experiment and simulation.
Since materials of interest in our current study are non-magnetic, we can have µ for The time-averaged Poynting vector (energy flux density) is given by 1,7 : Thus the energy flux density of scattered field through a given surface with a normal unit vector n ! is given by: Therefore, the absorption, scattering and extinction cross sections can be determined by 8 : where Q is the dissipative loss density, the integral volume V is the whole physical region

Parametric modeling of optical function of non-magnetic crystallographic lattice
One of the keys for computation with model described above is to determine parameters set for chiral nanostructures, which might not be always available.
S26 Supplementary Fig. 2a shows a general self-consistent approach to determine ξ , ε ( ) of a non-magnetic chiral material ( µ = 1 ) from experimental results. This approach is based on the fact that analytical solution of absorption and CD spectrum for nanoparticles with small size and spherical morphology can be available to allow comparison with related experimental results for parameter fitting. For example, for a small sized spherical particle, theoretical absorption cross section can be given by 5,9 : where B = V c ε m is a constant (c is the speed of light and V is the volume of nanoparticle). The dielectric function of chiral nanoparticles ε r , is a complex number given by ε r = ε 1 + iε 2 . The imaginary part can be described by a modified Tauc-Lorentz model (that is, combination of Tauc joint density of states with the Lorentz oscillator) 10,11 . For only a single inter-band transition is considered: where e t is the demarcation energy between the Urbach tail transition and the inter-band transition, e u is the Urbach width describing absorption tail associated with inter-band S27 transition, e 0 is the resonant energy of chiral nanoparticles, F and Γ are Lorentz oscillator amplitude and width, respectively. C 1 is a fitting parameter to ensure continuity of dielectric function at E = e t . This modified Tauc-Lorentz model can be extended for the scenario of multiple oscillators, with the example discussed below.
Once the imaginary part of dielectric function can be available, the real portion ε 1 can be acquired based on Kramers-Kronig relation 1,12 : where β denotes the principal value of the integral. By comparing experimental data acquired from small sized spherical nanoparticles with analytical solution of σ abs , material-dependent dielectric function parameter ε r can thus be determined.
In addition to the absorption, the CD response of a very small size chiral nanoparticles with spherical shape in a dielectric matrix ( ε m ) can be expressed as 5,9 : CD = CD ξ + CD ε .
The CD ξ and CD ε can be determined by: S28 where A = 8ω E !" 0 2 ε m 3/2 R 3 is a constant and R is the radius of nanoparticle. Here, we can setup a multi-oscillator model for fitting: where a j and Γ j are the oscillator strength and the damping factor, respectively.
Therefore, by comparing theoretical CD with experimental result of small sized chiral nanoparticle, chiral parameter, ξ can be therefore determined.
The parametric model as described above can be used to determine materialdependent parameters of optical function in a general manner, and the key is to utilize experimental data acquired from the spherical nanoparticles with very small size in order to improve fitting accuracy. We have used HgS as an example to illustrate our approach step-by-step to determine full set of parameters for computation of nanoparticles with achiral and chiral shape in current study, and we have also compared results from our approach with other techniques in order to substantiate our parametric modeling method in Supplementary Fig. 2. In order to validate our approach, we have synthesized small sized α-HgS nanoparticles with quasi-spherical morphology, and measured their absorption and CD spectra to compare with analytical expressions (see Supplementary   Fig. 2b and 2c). According to Tauc-Lorentz model, the fitting imaginary part of dielectric function of HgS is modeled by two oscillators and expressed as following: S29 The optimized fitting parameters from the Tauc-Lorentz model are summarized in Supplementary Table 1. We also present our fitted dielectric function ε 1 and ε 2 in Supplementary Fig. 3d. We would like to point out that our fitted dielectric function agrees well with previous experimental values of cinnabar and parameters from DFT calculation 9,13 . Furthermore, we have presented fitted imaginary and real parts of chiral parameters of HgS in Supplementary Fig. 2e, which are also in the same range of values from previous literatures 9 . All these agreements together validate our parametric modeling method.

Comparison between simulation and experimental results
Given the full set of parameters acquired in Supplementary Eqs. 12   However, in order to reduce computation cost for materials parametric fitting and simulation, our current work use isotropic dielectric function and chiral parameters, ξ , ε , µ ( ) . Even though such simplification with isotropic fitting parameters has been extensive used in prior literature, future improvement of simulation by developing a full S31 set of anisotropic fitting parameters should be necessary in order to fully reproduce fine features of chiroptical response in experiment.
Deionized water (H 2 O) was acquired by using Barnstead NANOPure water purification system, having a resistivity of 18.3 MΩ-cm. All aqueous solutions were prepared by using the deionized water.