Plasmonic Spherical Heterodimers: Reversal of Optical Binding Force Based on the Forced Breaking of Symmetry

The stimulating connection between the reversal of near-field plasmonic binding force and the role of symmetry-breaking has not been investigated comprehensively in the literature. In this work, the symmetry of spherical plasmonic heterodimer-setup is broken forcefully by shining the light from a specific side of the set-up instead of impinging it from the top. We demonstrate that for the forced symmetry-broken spherical heterodimer-configurations: reversal of lateral and longitudinal near-field binding force follow completely distinct mechanisms. Interestingly, the reversal of longitudinal binding force can be easily controlled either by changing the direction of light propagation or by varying their relative orientation. This simple process of controlling binding force may open a novel generic way of optical manipulation even with the heterodimers of other shapes. Though it is commonly believed that the reversal of near-field plasmonic binding force should naturally occur for the presence of bonding and anti-bonding modes or at least for the Fano resonance (and plasmonic forces mostly arise from the surface force), our study based on Lorentz-force dynamics suggests notably opposite proposals for the aforementioned cases. Observations in this article can be very useful for improved sensors, particle clustering and aggregation.


S2. Lateral optical force on Ag-Au and Au-Au heterodimers for on-axis configuration (a) Parallel Polarization: No reversal of lateral binding force for Au-Ag and Au-Au on-axis heterodimers
At first, we consider two on-axis [ȹ = 0] Ag-Au and Au-Au hetero-dimer set-ups of 100 nm and 50 nm with inter particle distance of 20 nm [cf. Fig. 1s  When Au-Au heterodimers of 76 nm and 50 nm are placed at 4nm distance and are shined with parallel polarization, Fano resonance takes place (cf. the case-2 in [1]). Similarly, for Ag-Au heterodimers such Fano resonance has been demonstrated in [2]. We have observed that even for such cases no reversal of the optical binding force occurs when the light polarization is parallel to dimer axis.
Moreover, a plasmonic "heterodimer" structure is expected to support both bonding and antibonding plasmon modes at the same time for both the transverse and longitudinal polarization due to its broken symmetry [3]. It is notable that for the parallel polarization, both the bonding and anti-bonding mode arises for the hetero-dimers [3]; but we have observed no reversal of optical binding force for Ag-Au or Au-Au or Ag-Ag hetero-dimers. As a result, in general, the reversal of lateral optical binding force for hetero-dimers (which will be shown in the next sub section) cannot be explained based on the idea of bonding and anti-bonding mode [4], too.
So, the important conclusion is that although reversal of optical binding force occurs for nano rods or other shapes due to Fano resonance [5,6], Fano resonance is in general not the reason of the reversal of optical binding force. So, particle size/shape, material and light polarizations are indeed important factors for such binding force reversal. Again, we consider two on-axis Ag-Au and Au-Au particles of 100 ad 50 nm with inter particle distance of 20 nm [cf. Fig. 1s (a), (b))] and perpendicular polarized light. The extinction cross section in Fig. 4(a) and (e) in the main article clearly reveal that both bonding resonance and anti-bonding resonance are occurring [3]. The mechanism of the force reversal of such hetero-dimers will be explained here shortly.
(1) At higher wavelength region (near 650 nm wavelength): it is observed that near the bonding resonance mode, reversal of the optical binding force, F Bind (x) = (F B (x) -F S (x) ) occurs at the wavelength 646 nm. Here F B(x) and F S(x) are the +x directed time averaged force on big and small particle respectively. At this specific wavelength two different hetero-dimers do not oscillate at lateral direction and the effective optical molecule does not experience any surface and bulk Lorentz force. The difference of the scattering part [cf. Eq (4) in the main article] or bulk part of the total Lorentz force [7,8] on a plasmonic object should describe the relative bulk force experienced by the optical molecule [Eq (5) given in main article]: Here; subscript (x), (B) and (S) represent: +x direction, bigger object and smaller object respectively. At the same time the difference of the gradient part [9] [which originates from induced surface charges; cf. Eq (3) in main article] of the total Lorentz force [7,8] on a plasmonic object should describe the relative surface force experienced by the optical molecule [Eq (6) given in main article]: At wavelength 646 nm, the right-hand sides of both Eqs (1s) and (2s) in this supplement are approximately zero. It should be noted that: It is also observed that the phase of the steady current and surface charges rapidly change just before and after this specific wavelength 646 nm as shown in Fig. 3s.   (2) At lower wavelength region (near 500 nm wavelength): for the transverse/perpendicular polarization case, in our Ag-Au, Au-Au and Ag-Ag heterodimer set-up, though spectral dip is observed at certain wavelengths, bonding dipole quadrupole (BDQ) pattern at the spectral minimum regions [1] is not found for the charge distribution of the objects (also cf. the discussion in [2] for perpendicular polarized light). As a result, this spectral minimum cannot be identified as Fano resonance/ Fano dip. We now consider a different idea [10]: the electric dipole moment of the objects to explain the reversal of binding force based on same and opposite electric charges. This idea of electric dipole moment [10] should be a more generalized idea than the electric polarizability (discussed in ref. [11]), as the overall size of the dimer set-up is higher than the dipolar limit in this work. The real part of electric dipole moment of an object is defined as [10]: It is observed that whenever the reversal of the lateral binding force occurs for transverse/perpendicular polarization, the real part of the electric dipole moment reverses its sign near the resonance for both the smaller object as shown in Fig 4s (a) and (c). On the other hand, it is observed that whenever no reversal of the lateral binding force occurs for longitudinal/parallel polarization, the real part of the electric dipole moment does not reverse their sign near the resonance for the smaller objects as shown in Fig 4s (b) and (d). So, the reversal of lateral binding force near this specific wavelength [near 500 nm as shown in Fig 4(b) and 4(f) in the main article] can better be explained based on the idea of induced same or opposite electric charges due to induced electric resonance [10] similar to the idea (reversal of the electric polarizability near resonance) proposed in ref. [11].  Fig. 1s (a)] and Au-Au heterodimers [for config. of Fig. 1s (b)]. All wavelengths are in meter (m) unit.
When the smaller particle is rotated keeping the bigger particle fixed, no reversal of the longitudinal binding force occurs for Ag-Ag and Au-Au off-axis [i.e. 70< ȹ <110] hetero-dimers similar to Ag-Au hetero-dimers discussed in main article. However, when the bigger object is rotated keeping the smaller one fixed for Au-Au and Ag-Ag hetero-dimers [cf. Fig. 1s (e) and (f) respectively], in Fig. 7s it is shown that for both polarizations of light, Ag-Ag off-axis heterodimers support the binding force reversal similar to Ag-Au hetero-dimers discussed in main article but Au-Au heterodimers do not support such reversal as shown in Fig. 8s [reason: the -y directed pushing force on the bigger object is always much higher than the smaller one and anti-bonding type resonance almost vanishes according to Fig. 8s (a),