Introduction

Chirality, referring to an object that cannot be superimposed on its mirror image by any combination of translations and rotations, plays a vital role in many areas of fundamental and applied science. Like our right and left hands, there exist in nature some mirror-twin molecules, known as enantiomers, whose handedness dictates their behaviors and functionalities, for instance, the pharmacological effects of chiral drugs on the human body1,2. Therefore, the ability to unambiguously detect molecular chirality is of crucial importance in various research disciplines, such as life science, molecular biology, and pharmacology as well as agricultural, chemical and pharmaceutical industries3,4.

Metallic and dielectric nanostructures and nanoparticles are attractive and promising nanoscale tools for the detection and analysis of the chirality (handedness) of molecules5,6,7 as well as generation of chiral light8,9,10. Their optical, plasmonic/magnetic properties, and resonance frequencies can be readily tailored by engineering the geometrical parameters to exhibit the desired near-field enhancement and far-field interference for realizing sensitive and reliable chirality sensing. In the past decade, numerous theoretical and experimental work has been carried out, such as planar nanoarrays/nanolattices11,12,13,14,15,16,17, metasurfaces/metamaterials18,19,20,21,22,23,24, and nanocubes/nanoassemblies25,26,27,28,29,30,31, for separation and discrimination of enantiomers at the specific resonant modes. These enantiomeric sensing platforms can be generally categorized as either chiral or achiral. The geometrically and structurally achiral ones are highly preferable because each building block and their spatial arrangement do not contribute to unwanted background signals so that the absolute measurement of chiral signals become feasible32,33. Although yielding a decent enhancement in chiroptical signals, most of these chirality sensing schemes relied merely on the excitation of either electric or magnetic resonances (typically one mode appears more prominent than the other)34,35,36,37,38. They underestimated the fact that the interaction of light with the chiral medium is significantly dependent on the excitation of both magnetic and electric modes39, i.e., a way to increase the magnetoelectric coupling introduced by the chirality terms appearing in the bi-isotropic constitutive relations40. Therefore, the chiroptical enhancements of the above-mentioned detection schemes are limited since the effect of the magnetoelectric coupling on the optical chirality is largely ignored and neglected. In addition, a thorough understanding of the underlying enhancement mechanism of the light-chiral matter interactions is still missing.

In this work, we present the use of a magnetoelectric nanoparticle (NP)41,42 composed of a metallic core and a dielectric shell to tackle the aforementioned obstacle and boost the interaction of light and chiral matter in a highly effective way. By judiciously varying the dimensions of the structurally achiral magnetoelectric NP decorated with a chiral monolayer, we systematically demonstrate that background-free chiroptical responses can be substantially enhanced through coherent multipole interactions at specific wavelengths, compared to non-magnetoelectric NPs, providing a qualitative understanding of the chiroptical enhancement mechanism with potential for detection of the chirality of monolayered thin film in an absolute measurement manner.

Results and discussion

We used Mie theory to investigate both scattering (absorption) efficiency and multipolar decomposition of NPs in the absence of a chiral shell layer, and then incorporated the eigenmodes and wavenumbers of electromagnetic waves propagating in a chiral medium into the Mie formulation to obtain chiroptical scattering (absorption) for the case of NPs capped with a chiral shell layer, as illustrated in Fig. 1. The NPs considered in this study are metallic/dielectric core-only and magnetoelectric core–shell nanospheres.

Fig. 1: Schematic of a magnetoelectric core–shell NP.
figure 1

The magnetoelectric core–shell NP capped with a chiral shell layer and encapsulated in a homogeneous host medium is illuminated with two circularly polarized (CP) waves of equal amplitude and opposite handedness, \({{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{L}}}}}}}^{{{{{{\bf{inc}}}}}}}\) and \({{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{R}}}}}}}^{{{{{{\bf{inc}}}}}}}\). \({{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{+}}}}}}}^{{{{{{\bf{sca}}}}}}}\) and \({{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{-}}}}}}}^{{{{{{\bf{sca}}}}}}}\) represent the total scattered fields in response to left- and right-handed CP [LCP (+) and RCP (−)] illuminations, respectively.

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Scattering efficiency and multipolar decomposition maps of NPs

Figure  2a, b, respectively, shows the dimensionless scattering efficiency Qsca maps—the scattering cross section Csca divided by the geometrical cross-sectional area as a function of wavelength and radius of the nanoparticle (NP) illuminated by a linearly polarized plane wave (Supplementary Notes 2 and 3 for the details of calculations)—of a gold (Au) and a silicon (Si) NP encapsulated in a lossless homogeneous background medium with refractive index nh = 1.33. The Qsca (Qabs) map allows for accurate estimation of the desired NP size at which the amount of scattered light (light absorbed by the scatterer) is maximum, giving guidelines for precisely tuning the resonance to the excitation wavelength range (see representative Qsca spectra (Supplementary Fig. S2) and the corresponding absorption efficiency Qabs maps (Supplementary Fig. S3) in Supplementary Note 4). One can observe that both metallic and dielectric NPs scatter (absorb) light differently and reveal extraordinary size-dependent scattering (absorption) efficiency. For Au NP, the Qsca is mainly dominated by the electric dipole (ED), and the magnetic multipoles are much less pronounced and only arise as the NP size increases, as illustrated by multipolar decomposition maps of Qsca in Fig. 2d. On the other hand, the Qsca of Si NP is mostly contributed by the distinct magnetic dipole (MD) and quadrupole (MQ) together with ED and electric quadrupole (EQ) in the form of a slow varying background, as shown in Fig. 2e. It should be noted that the influence of the high-order multipole terms, such as electric/magnetic octupole (EO/MO) and hexadecapole (EH/MH), on Qsca has to be taken into account as the scatterer size increases. Figure 2c shows the Qsca map of a magnetoelectric NP composed of a constant silver (Ag) core radius a and varying silicon shell thickness t. In contrast to the aforementioned Si NP, such a magnetoelectric Ag@Si NP not only preserves the same features of the magnetic multipoles as Si NP but also exhibits a quadratic-like (quartic-like) curve on the ED (EQ) map, which can be attributed to the excitation of electric dipole (quadrupole) mode at the core–shell interface, as shown in Fig. 2f. As a consequence, two intersection areas emerge (Fig. 2c), where the light-yellow straight lines shown in the MD and MQ subplots and the quadratic-like and quartic-like curves shown in the ED and EQ subplots of Fig. 2f are superimposed one another, as the silicon shell gets thicker. These multipole decomposition maps facilitate to explain the enhancement of the chiroptical responses based on the Qsca maps of the unperturbed NPs, and we will make extensive use of them in the following sections.

Fig. 2: Scattering efficiency and their corresponding multipolar decomposition maps.
figure 2

Scattering efficiency \({{{{{{\rm{Q}}}}}}}^{{{{{{\rm{sca}}}}}}}\) maps of a Au, b Si, and c Ag@Si NPs as a function of wavelength and core radius a (shell thickness t with a constant core radius a = 50 nm in c). d–f Corresponding multipolar decomposition maps of \({{{{{{\rm{Q}}}}}}}^{{{{{{\rm{sca}}}}}}}\), where the letters denote the multipoles as follows: ED (MD), electric (magnetic) dipole (n = 1); EQ (MQ), electric (magnetic) quadrupole (n = 2); EO (MO), electric (magnetic) octupole (n = 3); EH (MH), electric (magnetic) hexadecapole (n = 4). The single NP is embedded in a lossless homogeneous host medium. All plots in each row share the same color bar on the right.

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Circular differential scattering cross-section maps of chiral ligand-capped NPs

Let us consider the case that a one-nanometer-thick layer (tc = 1 nm) of a chiral substance regarded as a small disturbance adheres to the exterior surface of the unperturbed, achiral NP. The adsorbed chiral substance is characterized by an average refractive index \({n}_{c}=1.33+{10}^{-4}i\) and a chirality parameter κ. For a lossy medium, we have to ensure that the lossy requirement, \({{{{{{\rm{Im}}}}}}\left(\kappa \right)}^{2} \, < \, 2{{{{\mathrm{Re}}}}}\left({n}_{c}\right){{{{{\rm{Im}}}}}}\left({n}_{c}\right)\), is satisfied for given values of κ32. κ measures the degree of the handedness of the material and a change in the sign of κ means taking the mirror image of the material40. The presence of the outermost monolayer of the chiral substance (κ ≠ 0) introduce the magnetoelectric coupling between the electric and magnetic fields within the chiral monolayer. Circular differential scattering cross-section ∆σsca maps (in unit of nm2)—the difference between the scattering cross section σsca for left- and right-handed CP (LCP and RCP) incident light (see detailed discussions in Supplementary Note 3)—of Au and Si NPs capped with a chiral monolayer with κ = ±10−5i, typical values for the chirality parameter of chiral molecules15,43, are shown in the left and right columns of Fig. 3a, b, respectively. We have to emphasize that instead of a linearly polarized incident light, oppositely handed incident waves of equal amplitude are used as individual input states, namely \(\left({E}_{{iL}},{E}_{{iR}}\right)=\left({{{{\mathrm{1,0}}}}}\right)\) and (0, 1), to obtain two separate scattering cross sections \({{{{{{\rm{\sigma }}}}}}}_{\pm }^{{{{{{\rm{sca}}}}}}}\) where + and – signs in the subscript represent LCP and RCP illuminations, respectively, and then calculate their difference. It is apparent that the solitary Au NP is not a promising candidate for the chiral enhancer since the magnitude of the ∆σsca, proportional to the strength of the measured chiroptical responses, is negligibly small and can be ignored. As for Si NP, the first four highest (lowest) stripes of ∆σsca shown in the left (right) column of Fig. 3b, at the first sight, seem to follow the patterns of darker red and light-yellow stripes of Qsca (Fig. 2b). However, the ∆σsca appeared weaker in the corresponding orange-colored region of Qsca where the magnitude of Qsca is greater than that in the light-yellow stripes of Qsca. The reason is that ∆σsca is the differential quantity of σsca and only contributed by the specific combinations of superimposed multipoles—those spectrally overlapped electric and magnetic multipoles of the same order. The letters on ∆σsca maps mark the dominant multipoles that contribute to the chiroptical signals. The magnitude and sign of ∆σsca are proportional to the extent that the electric and magnetic multipoles of the same order [(ED,MD), (EQ,MQ) and (EO,MO) for example] are spectrally overlapped and dependent on the sign of Im(κ), respectively. This argument can be further verified by comparisons of the ∆σsca (Fig. 3c) of the perturbed Ag@Si NPs with the Qsca (Fig. 2c) of the unperturbed one. The global maximum and minimum of the chirality-dependent ∆σsca (Fig. 3c) occur at the spectral region where the peaks of EQ and MQ resonances, with a slow varying background of ED and MD modes, are maximally superimposed. In contrast, although the global extrema of Qsca (Fig. 2c) is found at 1.35 μm, the magnitude of the corresponding ∆σsca is noticeably weaker because of the partial overlap between the ED and MD resonances.

Fig. 3: Circular differential scattering cross-section maps of chiral ligand-capped NPs.
figure 3

Circular differential scattering cross-section \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\) maps of a Au, b Si, and c Ag@Si NPs coated with a chiral monolayer as a function of wavelength and core radius a (shell thickness t with a constant core radius a = 50 nm in c). These NPs are capped with a 1-nm-thick chiral layer (represented by a green-colored shell, \({t}_{c}=1\;{{{{{\rm{nm}}}}}}\)) with the chirality parameter \({{{{{\rm{\kappa }}}}}}=\mp {10}^{-5}i\) represented on the left and right columns, respectively. The letters on the figures mark the dominant multipoles that contribute to the chiroptical signals. All subplots in each row share the same color bar on the middle. Representative \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\) spectra of chiral ligand-capped d Si and e Ag@Si NPs with \({{{{{\rm{\kappa }}}}}}={10}^{-5}i\) for different values of a in d and [a, t] in e. Comparisons of analytically calculated and numerically simulated \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\) spectra of chiral ligand-capped f Si and g Ag@Si NPs with \({t}_{c}=10\, {{{{{\rm{nm}}}}}}\), \({n}_{c}=1.33+{10}^{-4}i\) and \({{{{{\rm{\kappa }}}}}}=\mp {10}^{-3}i\).

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Since both the core-only/magnetoelectric core–shell NPs and background medium have no inherent chirality, the resulting chiroptical responses come entirely from the inclusion of the chiral substance surrounding NPs. Figure 3d, e displays the ∆σsca spectra of different sizes of Si and Ag@Si NPs capped with a chiral monolayer with κ = 10−5 i, respectively. In order to maintain the same surface coverage of chiral molecules for fair comparisons, the radius of Si NP is set to be equal to the sum of the core radius and shell thickness of Ag@Si NP. Evidently, the chiroptical responses are considerably enhanced by Ag@Si NP since the superimposition of EQ and MQ resonances is maximally achieved by varying silicon shell thickness. Please also refer to the corresponding circular differential absorption cross-section ∆σabs maps (Supplementary Fig. S4 in Supplementary Note 4). Based on the ratio of the slopes of the fitted lines, obtained from the linear regression analyses in the near-infrared spectral region (Supplementary Fig. S5 in Supplementary Note 4), Ag@Si NP offers as high as 10-fold (5-fold) chiroptical enhancement in ∆σsca (∆σabs) with respect to the core-only Si NP. In order to validate our analytical results, COMSOL Multiphysics, which allows users to modify the underlying constitutive relations of the chiral medium, is used to compute ∆σsca (∆σabs) of chiral ligand-capped Si and Ag@Si NPs, as shown in Fig. 3f, g (Supplementary Fig. S4f, S4g in Supplementary Note 4). The chiral parameters \({t}_{c}=10\, {{{{{\rm{nm}}}}}}\), \({n}_{c}=1.33+{10}^{-4}i\) and \({{{{{\rm{\kappa }}}}}}=\pm \!{10}^{-3}i\) are selected to facilitate the simulation. The wavelength-dependent refractive indices of silver and silicon are chosen from the tabulated data44,45. Our analytical calculations (solid curves) are in excellent agreements with numerical simulations (hollow circles). In addition, we also examined the chiroptical responses arising from purely real κ (see Supplementary Note 5 for further details). Both ∆σsca and ∆σabs are greatly enhanced for \(\kappa =\pm \!{10}^{-3}i\), making a strong contrast with those for \(\kappa =\pm \!{10}^{-3}\) that reveal a negligibly small amplitude at the overlapped quadrupole resonance of the magnetoelectric NP. As a result, the proposed magnetoelectric-assisted chirality sensing scheme can be used to absolutely detect Im(κ) at the magnetoelectric (EQ,MQ) resonance even though Re(κ) ≠ 0.

Detailed correlations between scattering (absorption) efficiency and circular differential scattering (absorption) cross sections

We now take a deeper look at the correlations between the Qsca (Qabs) and ∆σsca (∆σabs) spectra of Ag@Si NPs with a fixed core radius and different shell thickness in the absence and presence of a chiral monolayer, as depicted in Fig. 4. This thoughtful examination not only allows us to systematically predict the behaviors of ∆σsca (∆σabs) based on the known features of Qsca (Qabs) and their multipolar decomposition spectra, but also provides an insight into how the combined excitation of orthogonal multipolar modes of the same order coherently enhances the magnetoelectric coupling induced by the adsorbed chiral substance and effectively boosts chiroptical signals. In the case of [a, t] = [63, 126] shown in the left panel, both Qsca and Qabs (Fig. 4a, b) reach local maximum nearby λQ1 where both MQ and EQ modes are maximally superimposed. At the same time, the amplitudes of ∆σsca and ∆σabs at λQ1 go to maximum as well, as shown in Fig. 4c, d, respectively. As for [a, t] = [63, 128] shown in the right panel, an additional increase in the silicon shell thickness results in a noticeable red shift ∆λ of MQ modes (Fig. 4e and the insets in Fig. 4f) and thus a slight drop in the peak amplitudes of Qsca and Qabs near \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}3}={{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}1}+\triangle {{{{{\rm{\lambda }}}}}}\) (Fig. 4e, f), leading to a decrease in the amplitudes of ∆σsca and ∆σabs at λQ3, as shown in Fig. 4g, h, respectively. It should be pointed out that the electric multipoles, such as EQ and EO modes, are less shifted than the magnetic multipoles of the same order owing to the constant silver core radius. Similarly, the same concept holds for high-order multipole terms. The maximal spectral overlap of EO and MO modes occurs at λO3 (the inset in Fig. 4e, f) and hence causes an evident increase in the amplitudes of ∆σsca and ∆σabs at the same wavelength, as shown in Fig. 4g, h, respectively. Although Qsca at λO3 (Fig. 4e) has nearly equal amplitude as that close to λO1 (Fig. 4a), ∆σsca is apparently increased at λO3 (Fig. 4g) because of the maximal spectral overlap of MO and EO modes (see insets in Fig. 4a, e). Our findings indicate that the enhancement of chiroptical responses significantly relies on the combined excitation of magnetic and electric multipoles of the same order, the degree of their spectral overlap and the strength of individual multipoles. This claim can be confirmed by examining the global maximum peak of Qabs around 666 nm, where partially overlapped MH and EO modes predominate over others, but they are not the same order (Fig. 4b, f). Insets in Fig. 4c, g show the representative far-field radiation patterns (\(\left|{{{{{{\bf{E}}}}}}}_{{{\pm }}}^{{{{{{\rm{far}}}}}}}\right|\) shown in x–z plane, φ = 0) of the chiral ligand-capped Ag@Si NP in response to LCP (+) and RCP (–) illuminations for both negative and positive Im(κ) at λQ1 (left panel) and λO3 (right panel), respectively. The insets clearly exhibit that the magnitude of far-field radiation is dependent on the handedness of the adsorbed chiral molecules and the angular shape of the radiation patterns is determined by the dominant EQ + MQ (EO + MO) mixed with the slow varying background of ED + MD (EQ + MQ + ED + MD)46, as shown in Fig. 4c (Fig. 4g).

Fig. 4: Detailed correlations between Qsca (Qabs) and σsca (∆σabs) spectra.
figure 4

a, e Scattering and b, f absorption efficiency, Qsca and Qabs, spectra of Ag@Si NPs for [core radius a, shell thickness t] = [63, 126] and [63, 128] shown on the left and right panels, respectively. Corresponding multipolar decomposition spectra are plotted with different colors. Zoomed-in insets in a, e and b, f highlight EO, MO, and EQ, MQ modes, respectively. Circular differential c, g scattering and d, h absorption cross section, \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\) and \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\), spectra of Ag@Si NPs capped with a chiral monolayer (\({t}_{c}=1\, {{{{{\rm{nm}}}}}}\)) with \({{{{{\rm{\kappa }}}}}}={10}^{-5}i\). The insets in c, g show the representative far-field radiation patterns of the chiral ligand-capped Ag@Si NPs in response to LCP (+) and RCP (–) illuminations for both negative and positive \({{{{{\rm{Im}}}}}}\left({{{{{\rm{\kappa }}}}}}\right)\) at \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}1}\) (left column) and \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}3}\) (right column), respectively. d, h Relative contributions of nonchiral (\(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{core}}}}}}}^{{{{{{\rm{abs}}}}}}}\) and \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{shell}}}}}}}^{{{{{{\rm{abs}}}}}}}\)) and chiral (\(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\)) portions to the overall \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) are computed based on Eqs. (1) and (2). The total \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) is equal to the sum of three individual contributions. \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\) is close to zero at the \({{{{{\rm{\lambda }}}}}}\)-intercept points \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{C}}}}}}1}\) and \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{C}}}}}}3}\).

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Although the absorption cross sections \({{{{{{\rm{\sigma }}}}}}}_{\pm }^{{{{{{\rm{abs}}}}}}}\) for LCP (+) and RCP (−) illuminations can be derived from chiroptical Mie calculations (see Supplementary Note 3), this approach does not tell us how much each individual material domain contributes. Instead, we evaluated \({{{{{{\rm{\sigma }}}}}}}_{\pm }^{{{{{{\rm{abs}}}}}}}\) by the volume integral of the dissipated power density \({p}_{\pm }^{d}\), where \({p}_{\pm }^{d}=-{{{{{\mathrm{Re}}}}}}\left[\nabla \cdot {{{{{{\bf{S}}}}}}}_{\pm }\right]={p}_{\pm }^{{{{{{\rm{core}}}}}}}+{p}_{\pm }^{{{{{{\rm{shell}}}}}}}+{p}_{\pm }^{{{{{{\rm{chiral}}}}}}}={p}_{\pm }^{{{{{{\rm{nonchiral}}}}}}}+{p}_{\pm }^{{{{{{\rm{chiral}}}}}}}\), as follows

$${\sigma }_{\pm }^{{{{{{\rm{abs}}}}}}}\,{I}_{inc}= \int\!\!\iint {p}_{\pm }^{d}dV=\, {k}_{0}\left[\int\!\!\iint {{\mbox{Im}}}({n}_{j})\frac{{{{{\mathrm{Re}}}}}({n}_{j})}{{\eta }_{0}}{|{{{{{{\bf{E}}}}}}}_{\pm }^{j}|}^{2}d{V}_{j} \right. \\ \,\left. + \int\!\!\iint {{\mbox{Im}}}(\kappa ){{\mbox{Im}}}[{{{{{{\bf{E}}}}}}}_{\pm }^{{{{{{\rm{c}}}}}}}\cdot {{{{{{\bf{H}}}}}}}_{\pm }^{{{{{{\rm{c}}}}}}\,\ast }]d{V}_{{{{{{\rm{c}}}}}}}\right]$$
(1)
$$\varDelta {\sigma }^{{{{{{\rm{abs}}}}}}}\equiv {\sigma }_{+}^{{{{{{\rm{abs}}}}}}}-{\sigma }_{-}^{{{{{{\rm{abs}}}}}}}=\varDelta {\sigma }_{{{{{{\rm{core}}}}}}}^{{{{{{\rm{abs}}}}}}}+\varDelta {\sigma }_{{{{{{\rm{shell}}}}}}}^{{{{{{\rm{abs}}}}}}}+\varDelta {\sigma }_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}$$
(2)
$$\varDelta {p}^{d}\equiv {p}_{+}^{d}-{p}_{-}^{d}=\varDelta {p}^{{{{{{\rm{core}}}}}}}+\varDelta {p}^{{{{{{\rm{shell}}}}}}}+\varDelta {p}^{{{{{{\rm{chiral}}}}}}}$$
(3)

where \({{{{\mathrm{Re}}}}}\left[\nabla \cdot {{{{{{\bf{S}}}}}}}_{\pm }\right]\) is the time-averaged power flow density, \({I}_{{inc}}\) the incident intensity, \({k}_{0}\) (\({\eta }_{0}\)) the free-space wavenumber (characteristic impedance), \({n}_{j}\) (\({{{{{{\bf{E}}}}}}}_{\pm }^{j}\)) the refractive index (electric field) at the volume element \(d{V}_{j}\) inside the domain \(j\) (j = core, shell or chiral shell label by \({{{{{\rm{c}}}}}}\)), and \({{{{{{\bf{E}}}}}}}_{\pm }^{{{{{{\rm{c}}}}}}}\) (\({{{{{{\bf{H}}}}}}}_{\pm }^{{{{{{\rm{c}}}}}}}\)) the electric (magnetic) field at the volume element \(d{V}_{{{{{{\rm{c}}}}}}}\) inside the chiral shell. Dissipated power density \({p}_{\pm }^{d}\) consists of \({p}_{\pm }^{{{{{{\rm{nonchiral}}}}}}}\) and \({p}_{\pm }^{{{{{{\rm{chiral}}}}}}}\) originating from three separate lossy sources, respectively: \({{{{{\rm{Im}}}}}}\left({n}_{{{{{{\rm{core}}}}}}}\right)\), \({{{{{\rm{Im}}}}}}\left({n}_{{{{{{\rm{shell}}}}}}}\right)\), and [\({{{{{\rm{Im}}}}}}\left({n}_{{{{{{\rm{c}}}}}}}\right)\) and \({{{{{\rm{Im}}}}}}\left(\kappa \right)\)], as indicated in Eq. (1). As a consequence, the total \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) is equal to the sum of three individual contributions, as expressed in Eq. (2). Relative contributions of nonchiral (\(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{nonchiral}}}}}}}^{{{{{{\rm{abs}}}}}}}=\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{core}}}}}}}^{{{{{{\rm{abs}}}}}}}+\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{shell}}}}}}}^{{{{{{\rm{abs}}}}}}}\)) and chiral (\(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\)) portions to the overall \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) are plotted with different colors, as illustrated in Fig. 4d, h. It is obvious that both \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\) and \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{nonchiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\) are only visible at the specific peaks of \({{{{{{\rm{Q}}}}}}}^{{{{{{\rm{sca}}}}}}}\) and \({{{{{{\rm{Q}}}}}}}^{{{{{{\rm{abs}}}}}}}\) where the electric and magnetic multipoles of the same order are superimposed. \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\) increases in much the same way as \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{nonchiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\), but is proportional to the sum of two volume integrals: (1) the product of \({{{{{\rm{Im}}}}}}\left({n}_{{{{{{\rm{c}}}}}}}\right)\) and the circular differential near-field intensity defined as \(\triangle {\left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{c}}}}}}}\right|}^{2}\equiv \frac{{{{{\mathrm{Re}}}}}\left({n}_{{{{{{\rm{c}}}}}}}\right)}{{\eta }_{0}}\left[{\left|{{{{{{\bf{E}}}}}}}_{+}^{{{{{{\rm{c}}}}}}}\right|}^{2}-{\left|{{{{{{\bf{E}}}}}}}_{-}^{{{{{{\rm{c}}}}}}}\right|}^{2}\right]\) and (2) the product of \({{{{{\rm{Im}}}}}}\left(\kappa \right)\) and the circular differential optical chirality density defined as \(\triangle {{{{{\rm{Im}}}}}}\left[{{{{{{\bf{E}}}}}}}^{{{{{{\rm{c}}}}}}}\cdot {{{{{{{\bf{H}}}}}}}^{{{{{{\rm{c}}}}}}}}^{* }\right]\equiv {{{{{\rm{Im}}}}}}\left[{{{{{{\bf{E}}}}}}}_{+}^{{{{{{\rm{c}}}}}}}\cdot {{{{{{{\bf{H}}}}}}}_{+}^{{{{{{\rm{c}}}}}}}}^{* }\right]-{{{{{\rm{Im}}}}}}\left[{{{{{{\bf{E}}}}}}}_{-}^{{{{{{\rm{c}}}}}}}\cdot {{{{{{{\bf{H}}}}}}}_{-}^{{{{{{\rm{c}}}}}}}}^{* }\right]\) (see Supplementary Note 6 for more details). Since the former contribution is infinitesimally small, \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\) is primarily contributed by the latter. In other words, the coherent multipolar amplification of chiroptical responses are entirely governed by the enhancement of \(\triangle {{{{{\rm{Im}}}}}}\left[{{{{{{\bf{E}}}}}}}^{{{{{{\rm{c}}}}}}}\cdot {{{{{{{\bf{H}}}}}}}^{{{{{{\rm{c}}}}}}}}^{* }\right]\) rather than by that of \(\triangle {\left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{c}}}}}}}\right|}^{2}\). For verification purpose, numerically simulated \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{core}}}}}}}^{{{{{{\rm{abs}}}}}}}\), \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{shell}}}}}}}^{{{{{{\rm{abs}}}}}}}\) and \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\) are in excellent agreements with our analytical calculations (see Supplementary Note 7 for further details).

Spatial distributions of chirality-dependent circular differential power dissipation density and far-field radiation patterns

With Eqs. (1) and (3), we can calculate spatial distributions of circular differential power dissipation density \(\triangle {p}^{d}\) of chiral ligand-capped Ag@Si NPs. The spatially chirality-dependent \(\triangle {p}^{d}\) at \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}1}\) and \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}3}\) are illustrated in Fig. 5a, b, respectively. Within the chiral monolayer region, \(\triangle {p}^{d}\)(\({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}1}\)) appears uniformly distributed in all angular directions except for \({{{{{\rm{\theta }}}}}}={90}^{^\circ }\) and \({270}^{^\circ }\), whereas \(\triangle {p}^{d}\)(\({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}3}\)) is much denser in the backward than forward direction, with an obvious achiral contribution from the silver core. For comparison purposes, we further examined reference and transition \(\triangle {p}^{d}\) plots, presenting the evolution of chiroptical signals and spatial distributions of chirality-dependent \(\triangle {p}^{d}\) at and between magnetoelectric resonances (see Supplementary Note 8). Since \(\triangle {{{{{{\rm{\sigma }}}}}}}_{{{{{{\rm{chiral}}}}}}}^{{{{{{\rm{abs}}}}}}}\) makes the dominant contribution to \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) (see Fig. 4d, h), the angular contours of \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}}\right)\) and \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}}\right)\), namely those spatial \(\triangle {p}^{d}\) within the chiral monolayer region denoted as \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\equiv {p}_{+}^{{{{{{\rm{chiral}}}}}}}-{p}_{-}^{{{{{{\rm{chiral}}}}}}}\) at the wavelengths of the overlapped quadrupole and octupole resonances, for various [a, t] and \({{{{{\rm{Im}}}}}}\left({{{{{\rm{\kappa }}}}}}\right)\gtrless 0\) are investigated and represented with different colors and line styles, as shown in Fig. 5c, d, respectively. The number of the lobes of \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\) is closely related to the spatial distributions of \(\triangle {{{{{\rm{Im}}}}}}\left[{{{{{{\bf{E}}}}}}}^{{{{{{\rm{c}}}}}}}\cdot {{{{{{{\bf{H}}}}}}}^{{{{{{\rm{c}}}}}}}}^{* }\right]\) at the excited quadrupole and octupole modes. Moreover, \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}}\right)\) is susceptible to the change in the silicon shell thickness, whereas \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}}\right)\) almost remains unaltered. In such representation of radar charts, it should be stressed that \(\triangle {p}^{{{{{{\rm{chiral}}}}}}} \, > \, 0\) (\(\triangle {p}^{{{{{{\rm{chiral}}}}}}} \, < \, 0\)) is situated outside (inside) of the black dashed circle on which \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}=0\) owing to \({{{{{\rm{Im}}}}}}\left({{{{{\rm{\kappa }}}}}}\right)=0\). For any given configuration of [a, t], the resultant \(\triangle {p}^{{{{{{\rm{chiral}}}}}}}\) have equal relative distance to the black dashed circle in the radial direction as long as the value of \(\left|{{{{{\rm{Im}}}}}}\left({{{{{\rm{\kappa }}}}}}\right)\right|\) remains the same. The same scenario holds for circular differential far-field radiation pattern defined as \(\triangle \left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{far}}}}}}}\right|\equiv \left|{{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{+}}}}}}}^{{{{{{\rm{far}}}}}}}\right|-\left|{{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{-}}}}}}}^{{{{{{\rm{far}}}}}}}\right|\). The corresponding chirality-dependent \(\triangle \left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{far}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}}\right)\right|\) and \(\triangle \left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{far}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}}\right)\right|\) are shown in Fig. 5e, f, respectively. Compared with \(\triangle \left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{far}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}}\right)\right|\), \(\triangle \left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{far}}}}}}}\left({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}}\right)\right|\) is less insensitive to variation of the NP’s structure and reveals a higher scattering amplitude. In a word, the quadrupole modes offered by Ag@Si NP are sensitive to changes in the molecular chirality while remaining robust to variation of the structure parameters (such as inconsistent size and/or shape distortion, mainly cased by imperfections during the fabrication and synthesis process of Ag@Si NPs47). As a result, the use of maximally overlapped EQ and MQ resonances is critical importance for the achiral magnetoelectric NPs to efficiently collect chiroptical signals using the dark-field configuration6 and to achieve reliable, robust and sensitive detection and identification of adsorbed enantiomeric molecules in the NIR spectral region. When it comes to total-field measurement, the circular differential extinction cross section \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{ext}}}}}}}\) is usually used as a measure of the amount of the adsorbed chiral substance in the experiment. In this work, however, we demonstrated that circular dichroism (CD) signal, proportional to the magnitude of \({{{{{\rm{Im}}}}}}\left(\kappa \right)\), can be absolutely detected regardless of the presence or absence of \({{{{\mathrm{Re}}}}}\left(\kappa \right)\) through the measurement of chiroptical scattering at the magnetoelectric (EQ,MQ) resonance where both chirality-dependent \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\) and \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) are substantially enhanced. We also realized that \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{abs}}}}}}}\) is a bit obscure to deal with since it is involved with both nonchiral and chiral terms, as compared to \(\triangle {{{{{{\rm{\sigma }}}}}}}^{{{{{{\rm{sca}}}}}}}\). So far there have not been any relevant experimental investigations. However, ref. 6 outlined a feasible detection configuration to measure such chiroptical signals.

Fig. 5: Spatial distributions of chirality-dependent ∆pd and ∆|Efar|.
figure 5

Spatial distributions of circular differential power dissipation density \(\triangle {p}^{d}\) of chiral ligand-capped Ag@Si NPs at a \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}1}\) and b \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}3}\). Ag@Si NPs are coated with a chiral monolayer (\({t}_{c}=1\,{{{{{\rm{nm}}}}}}\)) with \({{{{{\rm{\kappa }}}}}}=\mp {10}^{-5}i\) represented on the left and right columns, respectively. The angular contours of the chirality-dependent \(\triangle {p}^{d}\) within the chiral monolayer region (\({\triangle p}^{{{{{{\rm{chiral}}}}}}}\equiv {p}_{+}^{{{{{{\rm{chiral}}}}}}}-{p}_{-}^{{{{{{\rm{chiral}}}}}}}\propto {{{{{\rm{Im}}}}}}\left(\kappa \right)\triangle {{{{{\rm{Im}}}}}}\left[{{{{{{\bf{E}}}}}}}^{{{{{{\rm{c}}}}}}}\cdot {{{{{{\bf{H}}}}}}}^{{{{{{\rm{c}}}}}}* }\right]\) shown in x–z plane, \({{{{{\rm{\varphi }}}}}}=0\)) at c \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}}\) and d \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}}\). Different configurations of [a, t] and negative and positive \({{{{{\rm{Im}}}}}}\left({{{{{\rm{\kappa }}}}}}\right)\) are expressed with different colors and represented by the solid and dashed linens, respectively. The corresponding chirality-dependent circular differential far-field radiation pattern (\(\triangle \left|{{{{{{\bf{E}}}}}}}^{{{{{{\rm{far}}}}}}}\right|\equiv \left|{{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{+}}}}}}}^{{{{{{\rm{far}}}}}}}\right|-\left|{{{{{{\bf{E}}}}}}}_{{{{{{\boldsymbol{-}}}}}}}^{{{{{{\rm{far}}}}}}}\right|\) shown in x–z plane, \({{{{{\rm{\varphi }}}}}}=0\)) at e \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{Q}}}}}}}\) and f \({{{{{{\rm{\lambda }}}}}}}_{{{{{{\rm{O}}}}}}}\). The regions of ∆pchiral = 0 and ∆|Efar| = 0 are denoted by the black dashed circle on which Im(κ) = 0.

Full size image

In summary, we investigated scattering (absorption) efficiency maps of core-only and Ag@Si core–shell NPs and their corresponding multipolar decomposition maps through analytically solving Mie problems. We then incorporated the eigenmodes and wavenumbers of electromagnetic waves propagating in a chiral medium into Mie theory and obtained exact chiroptical Mie scattering (absorption) solutions for the case of NPs capped with a chiral monolayer. Having conclusive analytical evidence validated by numerical simulations, we confirmed that the magnitude of chiroptical responses is governed by the degree of spectral overlap of electric and magnetic multipoles of the same order and the strength of individual multipoles. We finally derived rigorous expressions for separation of relative contributions of the nonchiral and chiral parts of circular differential absorption, and analyzed the chirality-dependent circular differential power dissipation densities and far-field radiation patterns at different multipolar modes. Our findings pave the way for future research that fully exploits achiral magnetoelectric systems48 to coherently enhance the background-free chiroptical signals in the near-infrared wavelength regime. Our reported results might stimulate fundamental and applied research in different promising applications such as enhanced enantioselective spectroscopy, spin-controlled chiral spectrometry and generation of chiral light.

Methods

Eigencharacteristics of electromagnetic waves propagating in a chiral medium

The chiral medium can be characterized in terms of the bi-isotropic constitutive relations, relating the electric and magnetic field strengths (E and H) to the electric and magnetic flux densities (D and B), written in the time-harmonic convention \({e}^{-i\omega t}\) as40

$${{{{{\bf{D}}}}}}={\varepsilon }_{0}{\varepsilon }_{r}{{{{{\bf{E}}}}}}+i(\kappa /c){{{{{\bf{H}}}}}},\,{{{{{\bf{B}}}}}}={\mu }_{0}{\mu }_{r}{{{{{\bf{H}}}}}}-i(\kappa /c){{{{{\bf{E}}}}}}$$
(4)

where \({\varepsilon }_{0}\) is the permittivity, \({\mu }_{0}\) the permeability, and \(c\) the speed of light of free space. \({\varepsilon }_{r}\) is the relative permittivity, \({\mu }_{r}\) the relative permeability, and \(\kappa\) the chirality parameter of the chiral substance under study. To find the propagation characteristics of electromagnetic waves in such a medium, the time-harmonic Maxwell’s equations are compactly expressed in terms of two \(6\times 6\) Hermitian matrices and a single \(6\times 1\) column vector49, \({{{{{\mathscr{V}}}}}}{{{{{\boldsymbol{=}}}}}}{\left[\begin{array}{cc}{{{{{\bf{E}}}}}} & {\eta }_{0}{{{{{\bf{H}}}}}}\end{array}\right]}^{T}\), given by

$$\left[\begin{array}{cc}0 & i\nabla \times \\ -i\nabla \times & 0\end{array}\right]{{{{{\mathscr{V}}}}}}={k}_{0}\left[\begin{array}{cc}{\varepsilon }_{r} & i\kappa \\ -i\kappa & {\mu }_{r}\end{array}\right]{{{{{\mathscr{V}}}}}}$$
(5)

where the superscript T denotes transpose and \({\eta }_{0}\) (\({k}_{0}\)) is the free-space characteristic impedance (wavenumber). In a homogeneous chiral medium, the complex-valued \({{{{{\bf{E}}}}}}\) and \({{{{{\bf{H}}}}}}\) fields can be assumed to have a spatial dependence \(\exp \left(i{{{{{\bf{k}}}}}}{{{{{\boldsymbol{\cdot }}}}}}{{{{{\bf{r}}}}}}\right)\), where the wavevector \({{{{{\bf{k}}}}}}\, {{{{{\boldsymbol{=}}}}}}\, k\hat{{{{{{\bf{k}}}}}}}\) contains wavenumber \(k\) (i.e., the magnitude of \({{{{{\bf{k}}}}}}\)) and unit direction vector \(\hat{{{{{{\bf{k}}}}}}}\). It turns out that the solution of Eq. (5) becomes solving an eigenvalue problem as

$${\left[\begin{array}{cc}{\varepsilon }_{r} & i\kappa \\ -i\kappa & {\mu }_{r}\end{array}\right]}^{-1}\left[\begin{array}{cc}0 & -\hat{{{{{{\bf{k}}}}}}}\times \\ \hat{{{{{{\bf{k}}}}}}}\times & 0\end{array}\right]{{{{{{\mathscr{V}}}}}}}_{m}=({k}_{0}/{k}_{m}){{{{{{\mathscr{V}}}}}}}\!\!_{m}$$
(6)

The resultant \(6\times 6\) matrix has six eigenvalues \({k}_{0}/{k}_{m}\) and associated eigenvectors \({{{{{{\mathscr{V}}}}}}}\!\!_{m}\). The refractive index \({n}_{m}\) for each of eigenmodes is given by the inverse of the eigenvalues, and the wavenumber \({k}_{m}\) in general depends on the direction of the propagation. If the light is assumed propagating along the radial direction, four of \({k}_{m}\) come in two pairs of opposite sign and equal magnitude, representing the wavenumbers of the medium for circularly polarized (CP) waves of opposite handedness propagating along the positive and negative radial directions32. Thanks to Hermitian properties, their corresponding eigenvectors play the role of the circular basis vectors for the oppositely handed CP electric and magnetic fields propagating in either direction. The remaining two eigenvalues are zero, corresponding to the static fields with longitudinal eigenvectors \({\left[\begin{array}{cc}\hat{{{{{{\bf{r}}}}}}} & 0\end{array}\right]}^{T}\) and \({\left[\begin{array}{cc}0 & \hat{{{{{{\bf{r}}}}}}}\end{array}\right]}^{T}\)50. Therefore, the allowed solutions of the electromagnetic waves in the chiral medium are four distinct eigenmodes, summarized in Supplementary Table 1 (Supplementary Note 1). Knowledge of the eigenmodes propagating in the chiral medium and their orthonormal bases allows for the formulation of the representation of chiroptical fields in terms of vector spherical harmonics and calculations of cross sections in response to chiral perturbation51,52 (see Supplementary Notes 2 and 3 for detailed discussions).