# Fermi surface studies of a non-trivial topological compound YSi

Vikas Saini  Souvik Sasmal  Ruta Kulkarni  Bahadur Singh  A. Thamizhavel
<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400005, India.

###### Abstract

The Fermi surface properties of a nontrivial system YSi is investigated by de
Haas-van Alphen (dHvA) oscillation measurements combined with the first-
principle calculations. Three main frequencies ($\alpha$, $\beta$, $\gamma$)
are probed up to $14$ T magnetic field in dHvA oscillations. The
$\alpha$-branch corresponding to $21$ T frequency possesses non-trivial
topological character with $\pi$ Berry phase and a linear dispersion along
$\Gamma$ to $Z$ direction with a small effective mass of $0.069~{}m_{e}$ with
second-lowest Landau-level up to $14$ T. For $B~{}\parallel$ [010] direction,
the 295 T frequency exhibits non-trivial $2D$ character with $1.24\pi$ Berry
phase and a high Fermi velocity of $6.7\times 10^{5}$ ms-1. The band structure
calculations reveal multiple nodal crossings in the vicinity of Fermi energy
$E_{f}$ without spin-orbit coupling (SOC). Inclusion of SOC opens a small gap
in the nodal crossings and results in nonsymmorphic symmetry enforced Dirac
points at some high symmetry points, suggesting YSi to be a symmetry enforced
topological metal.

##  I Introduction

Topological semimetals have gained significant attention both on theoretical
and experimental fronts in recent years owing to their exotic properties. They
exhibit symmetry protected bulk and surface states which results novel
phenomena and transport properties with tremendous potential for applications
in quantum technologies and energy sciences Bansil  _et al._ (2016); Hasan and
Kane (2010); Qi and Zhang (2011); Vergniory  _et al._ (2019); Hu  _et al._
(2019); Fu and Kane (2007). Topological semimetals can be classified based on
the dimensionality and degeneracy of the crossing points. In particular,
topological Dirac semimetal has zero-dimensional four-fold degenerate crossing
points in the vicinity of Fermi energy ($E_{f}$). The low energy excitations
of the so-called Dirac particles follow linear energy dispersion around the
Dirac point and are the reasons for the observed exotic phenomenon in the
electrical transport measurements. The four-fold degeneracy of the Dirac point
is ensured by the presence of both the time-reversal $\mathcal{T}$ and
inversion $\mathcal{I}$ symmetries along with specific crystalline symmetries.
This four-fold degeneracy is lifted by breaking either inversion or time-
reversal symmetries, thus Dirac semimetals (DSM) transitions to Weyl
semimetals (WSM) with two-fold Weyl nodal crossings Singh  _et al._ (2012);
Wang  _et al._ (2013, 2012); Lv  _et al._ (2015). Cd3As2 and Na3Bi are known
prototype topological Dirac semimetals which was predicted theoretically and
realized in experiments. They exhibit ultra-high mobility, extremely high
magnetoresistance and show exotic phenomena such as chiral anomaly, quantum
hall effect, among others owing to symmetry protected Dirac band crossings.
The extensive studies of Shubnikov de-Haas(SdH) and de Haas-van Alphen(dHvA)
oscillations confirmed topological nontrivial nature of these compounds from
the Berry phase analysis and other unusual properties Wang  _et al._ (2013,
2012); Xu  _et al._ (2013); Zhang  _et al._ (2019, 2017); Kushwaha  _et al._
(2015); Chiu and Schnyder (2015); Liang  _et al._ (2015).

In view of finding new topological semimetals, we investigate Fermi surface
and topological properties of binary compound YSi by de Haas-van Alphen (dHvA)
oscillations and first-principles calculations. We synthesize high-quality
single crystals of YSi using the Czochralski method. From dHvA quantum
oscillations studies, YSi is found to have nontrivial Fermi pockets. The
first-principle calculations show that it has a rich nodal structure at the
Fermi level with symmetry protected band crossings. Three different branches
($\alpha,\beta$, and $\gamma$) are probed up to a magnetic field of $14$ T.
The nontrivial $\alpha$ pocket is observed along the three directions with a
subtle variation in the frequency implying a non-uniform cross-section of the
Fermi surface suggesting an anisotropic nature of the $\alpha$ pocket. On the
other hand, the other two pockets ($\beta$ and $\gamma$) are observed only
along $B~{}\parallel$ [010] direction suggesting a $2D$ nature of the Fermi
surface, where $\beta$ branch has the nontrivial character of topology. The
detailed band structure analysis shows that both the valence and conduction
bands participate in the formation of the Fermi surface with the hole and
electron pockets. The type-I nodal anti-band crossing along the $\Gamma-Z$
direction in the vicinity of Fermi energy leads to the nontrivial $\alpha$
pocket.

##  II Methods

From the binary phase diagram of Y and Si, it is obvious that YSi melts
congruently at 1845 ∘C Button  _et al._ (1990) and hence can be grown directly
from its melt. A tetra-arc furnace has been used to grow the single crystal of
YSi by the Czochralski method. High purity starting materials of Y (3N pure,
Alfa-Aesar) and Si (5N pure, Alfa-Aesar) were taken in the ratio of $1:1.05$
and repeatedly melted to make a homogenous polycrystalline ingot of about 8 to
10 g. A seed crystal was cut from this polycrystalline ingot to grow the
single crystal in a vacuum chamber filled with Ar gas. The polycrystalline
ingot was melted and the seed crystal was slowly inserted into the melt and
pulled very rapidly to start with. Once the steady-state condition is achieved
the pulling rate was maintained at $10$ mm/h. As grown pulled ingot had a
diameter $\approx 4$ mm and length $\approx 70$ mm and is shown in Fig. 1(a).
The composition analysis was performed using energy dispersive analysis by
x-rays (EDAX). A small portion of the crystal was crushed to fine powders and
subjected to room temperature powder x-ray diffraction (XRD) measurement in
PANalytical x-ray diffractometer equipped with a monochromatic
Cu-$K_{\rm\alpha}$ source with the wavelength $\lambda$ = 1.5406 Å. To confirm
the single-crystalline nature of the sample and to orient and cut the crystal
along the three principal crystallographic directions we have performed Laue
diffraction in the back reflection geometry. The oriented crystal was cut into
a rectangular bar shape using a spark erosion cutting machine. Magnetic
measurements were performed in a vibrating sample magnetometer (VSM) (PPMS,
Quantum Design, USA), down to $2$ K and in a magnetic field of $14$ T.

Band structure calculations were carried out with the projector augmented wave
(PAW) methodBlöchl (1994) within the density functional theory (DFT)Hohenberg
and Kohn (1964) framework as implemented in the Vienna ab initio simulation
package (VASP)Kresse and Joubert (1999); Kresse and Furthmüller (1996). The
exchange-correlation effects were considered with the generalized gradient
approximation (GGA) with the Perdew–Burke–Ernzerhof parameterizationPerdew
_et al._ (1996). The spin-orbit coupling (SOC) was included self-consistently.
An energy cut-off of 310 eV was used for the plane-wave basis set and a
$11\times 11\times 13$ $\Gamma$-centered k mesh was used for the bulk
Brillouin zone sampling. The Xcrysden program was used to visualize the Fermi
surfaceKokalj (1999). The robustness of results is further verified by
calculating electronic properties using the WIEN2K code which considers a
full-potential linearized augmented plane-wave formalism Blaha  _et al._
(2001). The quantum oscillations calculation were performed using the SKEAF
codeRourke and Julian (2012).

##  III Results and Discussion

###  III.1 Crystal Structure

Figure 1: (a) As grown single crystalline pulled ingot. (b) Room temperature
powder x-ray diffraction and structural refinement by Rietveld method. The
inset shows crystal structure of YSi with $b$ as the longest axis. The large
(yellow) and small (red) balls identify Y and Si atoms, respectively. (c),
(d), and (e) show the observed Laue diffraction pattern along the three
principal crystallographic planes (100), (010), and (001), respectively.

We start discussing the crystal structure of our grown single crystals of YSi.
The phase purity of the grown crystal was analysed by powder XRD measurement.
It has been reported that YSi crystallizes in the centrosymmetric orthorhombic
crystal structure with space group $Cmcm$ (#63) Parthe (1959). The unit cell
structure of YSi is shown in the inset of Fig. 1(b). The powder XRD pattern is
shown in Fig. 1(b) and it is evident that there are no discernible impurity
peaks in the entire $2\theta$ rage from 10 to 90 ∘. From the Rietveld analysis
using FULLPROF software Rodríguez-Carvajal (1993) we confirmed that this
compound crystallizes in the $Cmcm$ space group. The obtained lattice
parameter from the Rietveld analysis are $a=4.260$ Å, $b=10.530$ Å and
$c=3.830$ Å, which is in close agreement with the available data Parthe
(1959). The Y and Si atoms occupy $4c$ Wyckoff position with coordinates (0,
0.3598, 0.25) and (0, 0.0764, 0.25), respectively. The Laue diffraction
pattern attests good quality of the grown crystal as shown in Fig. 1(c), (d),
and (e) respectively for (100), (010), and (001) planes.

Figure 2: (a) Bulk Brillouin zone of orthorhombic YSi. The high-symmetry
points are marked. Bulk band structure of YSi (b) without and (c) with spin-
orbit coupling (SOC). (d) Orbital resolved band structure without SOC. Y $d$
and Si $p$ states are shown with gray and blue markers, respectively. (e)-(f)
Closeup of the bands along $\Gamma-Z$ direction in the area highlighted by
yellow rectangles in (b)-(c). The broken circles highlight the three crossing
bands. The crossing bands are gapped in the presence of SOC. (g)-(h) The
calculated individual Fermi pockets and (i) the Fermi surface of YSi.

###  III.2 Electronic structure

Figure 2(a) shows the bulk Brillouin zone for the primitive crystal structure
of YSi where high-symmetry points are marked explicitly. The calculated bulk
band structure of YSi along various high-symmetry directions without SOC is
shown in Fig. 2(b). It is seen to be metallic where various bands cross the
Fermi level. Importantly, many symmetry protected spinless band crossings are
found along the high-symmetry lines such as $R-Z$ and $T-Z$ at the Brillouin
zone boundaries as well at the generic $k-$ points. Along the $\Gamma-Z$
direction three bands cross, forming both the type-I and type-II nodal
crossings as shown in Fig. 2(e). The orbital resolved band structure shows
that these band crossings are composed by Y $d$ and Si $p$ orbitals (Fig.
2(d)). The structure with SOC is shown in Figs. 2(c) and (f). The various
nodal crossings at the generic $k$ points are gapped. However owing to the
presence of screw rotations $\\{C_{2z}|00\frac{1}{2}\\}$ and
$\\{C_{2y}|00\frac{1}{2}\\}$ and glide mirror $\\{M_{y}|00\frac{1}{2}\\}$, the
band crossings at $R$, $Z$, and $T$ points remain protected, referring high
symmetry point Dirac states.

The calculated bulk Fermi surface of YSi is shown in Fig. 2(i) whereas its
constituents individual pockets are illustrated in Figs. 2(g) and (h). Owing
to the multi-band crossings at the Fermi level, the Fermi surface is composed
of both the electron and hole bands. We mark a small pocket along the
$\Gamma-Z$ direction as $\alpha$ which is formed by bands highlighted in the
broken red circle in Fig. 2(f). On the other hand, the pocket along the $T-A1$
direction is identified as $\gamma$. Additionally, giant Fermi pockets are
enclosing $\Gamma$ and $Z$ points. The calculated quantum oscillations
frequencies are summarized in Table 1. It is found that a shift of $\sim+11$
meV in the Fermi level is essential to reproduce the experimentally observed
frequencies. The observed oscillations of the $\alpha$ pocket are observed in
all three directions and it carries the lowest frequency of $21$ T for
$B~{}\parallel~{}$[100]. $\beta$ pocket is observed only for B $||$ [010] and
has a frequency of $61$ T. Also, the $\gamma$ pocket was observed when B $||$
[010] with a high frequency of $295$ T. These oscillations are well captured
in our first-principles results (see below for more details).

###  III.3 de Haas-van Alphen quantum oscillations studies

The field dependence of the magnetization measurement measured in a VSM at 2
K, up to a field of 14 T, along the three principal crystallographic
directions is shown in Fig. 3(a), (b) and (c). Robust dHvA oscillations are
observed along all three directions. Long-period oscillations are observed for
$B~{}\parallel$ [100] and [001] directions while for [010] direction strong
oscillations with multiple frequencies are observed. At $2$ K the oscillations
begin to appear from $2$ T field and this corresponds to the magnetic length
$l_{\rm B}=\sqrt{\frac{\hbar}{eB}}$ $\approx$ 18 nm thus indicating a good
quality of the grown single crystal. As the temperature is increased, the
amplitudes of the dHvA oscillations tend to decrease and are not discernible
for a temperature greater than 21 K. The non-oscillatory background data was
subtracted to extract the dHvA oscillation frequency by fast Fourier transform
(FFT). The FFT spectra at $T=2$ K along the three principal crystallographic
directions are shown in Fig. 3(d),(e) and (f). When the magnetic field is
along the [100] direction, a single natural frequency at $21$ T has been
observed which is named as $\alpha$. For $B~{}\parallel$ [010] three natural
frequencies have been observed at $36$ T ($\alpha$), $61$ T ($\beta$) and
$295$ T ($\gamma$) with the second harmonics at $590$ T ($2\gamma$), while for
$B~{}\parallel$ [001] a single natural frequency at $34$ T ($\alpha$) and its
corresponding second harmonics was observed at $68$ T ($2\alpha$). It is to be
mentioned here that the $\alpha$-branch is observed along all the three
directions with a subtle anisotropy in the frequency thus representing a small
anisotropic $3D$ Fermi surface.

Figure 3: (a), (b), and (c) dHvA quantum oscillations observed along the three principal crystallographic directions in YSi at $T=2K$. (d), (e), and (f) The obtained FFT frequency spectrum. The $\alpha$ branch is observed along all the three directions while $\beta$ and $\gamma$ are observed only along the [010] direction. Table 1: Quantum parameters estimated from dHvA oscillations  B | $F_{exp}$ | $F_{cal}$ | $m^{*}_{exp}$ | $m^{*}_{cal}$ | $\tau_{q}$ | $\mu_{q}$ | $k_{\mathrm{F}}$ | $v_{\mathrm{F}}$ | $l_{q}$ | $n$
---|---|---|---|---|---|---|---|---|---|---
| (T) | (T) | $(m_{e})$ | $(m_{e})$ | $(10^{-13}sec)$ | $(cm^{2}V^{-1}s^{-1})$ | (Å${}^{-1})$ |  $(10^{5}$m/s) | (nm) |
[100] | 21 ($\alpha$) | 21.8 | 0.069(1) | 0.082 |  |  | 0.025 | 4.26 |  | 5.56 *$10^{17}cm^{-3}$
[010] | 295 ($\gamma$) | 294.7 | 0.162(2) | 0.147 | 2.2 | 2384 | 0.095 | 6.75 | 148.65 | 7.14 * $10^{12}cm^{-2}$
| 61 ($\beta$) | 71.5 | 0.097(2) | 0.162 | 1.14 | 2053 | 0.043 | 5.1 | 58.39 | 1.48*$10^{12}cm^{-2}$
[001] | 34 ($\alpha$) | 28.1 | 0.096(4) | 0.095 | 2.15 | 3927 | 0.032 | 3.85 | 82.77 | 1.109*$10^{18}cm^{-3}$

We have estimated the quantum parameters by analyzing the frequency spectrum
of the observed oscillations. The temperature dependence of the FFT amplitude
of the three natural frequencies for $B~{}\parallel$ [010] is shown in Fig.
4(a). The amplitude of the frequencies decreases as the temperature increases.
The $T$-dependence of the amplitude of the oscillation frequencies are shown
Fig. 4(a) and (b) and are fitted to thermal damping factor of Lifshitz-
Kosevich expression: $R_{T}=(X/sinhX)$, where $X=(\lambda Tm^{*}/H)$,
$\lambda=(2\pi^{2}k_{B}m_{e}/e\hbar)(=14.69$ T) and $m^{*}$ is the cyclotron
effective mass of the charge carriers which is expressed in units of free
electron mass $m_{e}$ Hu  _et al._ (2019); Matin  _et al._ (2018); Mondal  _et
al._ (2020) . The calculated effective masses for the various frequencies are
listed in Table 1. It is evident from the table that the effective masses of
charge carriers are very small for all the observed main frequencies
suggesting Dirac-like dispersion of bands and comparable to that of the gap-
less Dirac system Cd3As2 He  _et al._ (2014).

To understand the topological character of the charge carriers in YSi, we have
performed the Berry phase analysis of the dHvA oscillations using the
Lifshitz-Kosevich (L-K) formula Kumar  _et al._ (2017); Hu  _et al._ (2016) as
given below,

$\Delta
M\propto-B^{k}R_{T}R_{D}R_{s}sin\left[2\pi\left(\frac{F}{B}+\psi\right)\right],$
(1)

where $R_{T}$, $R_{D}$, and $R_{S}$ are the thermal, magnetic field and spin
damping factors and $\psi$ is the phase factor. The expression for $R_{T}$ has
already been defined while that for $R_{D}$ is $R_{D}=exp(-14.69m^{*}T_{\rm
D}/B)$ where $T_{\rm D}$ is the Dingle temperature and $R_{\rm S}$ = $cos(\pi
grm^{*}/2m_{0})$, where $\mathtt{g}$ is the Landé $\mathtt{g}$-factor and $r$
is the harmonic number. In Eqn. 1 $k=1/2$ for 3D Fermi surface while it is
zero for 2D Fermi surface. The phase factor $\psi$ is given by $\psi$=
[$(\frac{1}{2}-\frac{\Phi_{B}}{2\pi})-\delta$], where $\delta$ is the
additional phase factor which depends on the dimensionality of the Fermi
surface. $\delta=0$ for $2D$ Fermi surface and $\delta=\pm 1/8$ for $3D$ Fermi
surface where the $+$ sign corresponds to hole pocket and $-$ sign corresponds
to electron pocket.

Figure 4: (a) Temperature dependence of the FFT amplitude of the oscillation.
The FFT amplitude decreases with increasing temperature. (b), (c) Mass plot of
the frequencies mentioned along the three principal crystallographic
directions. The solid lines are the fits to the thermal damping factor of the
Lifshitz-Kosevich expression (see text for details).

For $B~{}\parallel$ [010] two additional frequencies are observed at 61 T and
295 T while these two frequencies are absent in the other two directions. This
indicates the anisotropic nature of the Fermi surface and these two
frequencies represents a $2D$ Fermi surface. Using the Onsager’s relation
$F=\frac{\hbar}{2\pi e}A_{\rm F}$, we have estimated the cross-sectional area
for $\gamma$ frequency as 0.028 Å-2. In order to estimate the Dingle
temperature $T_{\rm D}$ of the $\gamma$-branch, we have used the band-pass
filter to isolate the oscillations corresponding to the frequency 295 T and
used the $2D$ L-K expression in the high magnetic field region as shown in
Fig. 5(a). The reasonably good fit resulted in a Dingle temperature $T_{\rm
D}$ as 5.52 K. A similar estimate of the Dingle temperature was made for the
$\beta$ which resulted in $T_{\rm D}$ = 10.62 K. From the obtained values of
$m^{*}$, $T_{\rm D}$, we have estimated the Fermi wave vector $k_{\rm F}$,
Fermi velocity $v_{\rm F}$, quantum scattering $\tau_{\rm q}$, quantum
mobility $\mu_{\rm q}$ and the surface carrier density $n_{\rm 2D}$ as listed
in Table 1. The information about the Berry phase in YSi has been extracted by
plotting the Landau level (LL) fan diagram. We have assigned the LL-index
$n(=F/B+\psi)$ to the minima of quantum oscillations. For $B~{}\parallel$
[010] there are multiple frequencies in the quantum oscillation hence we have
used the band pass filtered oscillation to construct the LL-fan diagram which
is plotted as $n$ as a function of the inverse magnetic field as shown in Fig.
5(b) and (d). The plots are straight line and the slope corresponds to the
oscillation frequency while the intercept gives the phase factor $\psi$
mentioned in Eqn. 1. For the $\gamma$-branch, the intercept is $-0.12$, this
amounts to a Berry phase ($\Phi_{\rm B}$) of $1.24\pi$, which reveals the non-
trivial character of $\gamma$ pocket. Similarly, for the $\beta$-branch (61 T)
the intercept was estimated to be $-0.3$, this amounts to $\Phi_{\rm B}$ of
$1.6~{}\pi$ suggesting a trivial nature of this band and for the
$\alpha$-branch the intercept was $0.114$ and $-0.82$ for $B~{}\parallel$
[100] and $B~{}\parallel$ [001] directions (Fig. 5(e) and (f)), respectively
with $\Phi_{\rm B}$ = $\pi$ and $2.89\pi$ suggesting a non-trivial nature of
the Fermi surface. For $\alpha$-branch the second-lowest Landau level has been
achieved in a field of $14$ T that exhibits a small Fermi surface which is
smaller than the reported Fermi surfaces for Cd3As2 and ${\rm ZrSiS}$ systems
He  _et al._ (2014); Hu  _et al._ (2017).

Figure 5: (a) The band pass filtered dHvA oscillations of the $\gamma$-branch
measured at $T=2$ K for $B~{}\parallel$ [010], the solid line corresponds to
the L-K formula fitting. (b) LL-fan diagram corresponding to the
$\gamma$-branch. (c) L-K formula fitting of the filtered $\beta$-branch
oscillations, (d) the LL-fan diagram corresponding to $\beta$-branch. (e) and
(f) LL-fan diagram of the $\alpha$-branch along the [100] and [001]
directions.

##  IV Conclusion

We have grown the single crystal of YSi using the Czochralski method. The band
structure shows multiple nodal band crossings near the Fermi level. The Fermi
surface is formed by both the electron and hole bands, resulting in multi
Fermi pockets with both small and large areas. Importantly, our calculated
quantum oscillation frequencies match well with the observed frequencies. The
dHvA oscillation measurements performed along the three principal
crystallographic directions have revealed three Fermi pockets, namely,
$\alpha$, $\beta$, and $\gamma$, which are well seen in our first-principles
calculations. The $\alpha$ band obeys linear dispersion around the Fermi
energy which results in a very small effective mass $0.069~{}m_{e}$ for
$B~{}\parallel$ [100]-direction and in fact, it is lighter than other popular
Dirac and Weyl semimetals Du  _et al._ (2016); Gao  _et al._ (2017); Butcher
_et al._ (2019); Wu  _et al._ (2019); Singha  _et al._ (2017); Inamdar  _et
al._ (2013); Chen  _et al._ (2018). The estimated Berry phase for this branch
is $\pi$ which refers to a non-trivial topological character for this pocket
and is observed along all the three principal directions with subtle
anisotropy. Similarly, $\gamma$-branch also possess very light mass because of
linear band dispersion. The observed non-trivial Fermi pocket corresponding to
the $\gamma$-branch shows a $2D$ character with a quantum scattering time
$2.2~{}\times~{}10^{-13}$ s, which is of the same order as observed in ZrSiS
and PtBi2 systems Gao  _et al._ (2017); Hu  _et al._ (2017). The large quantum
mean free path and high quantum mobility signifies the suppression of
backscattering in YSi. These results make YSi is an interesting topological
material where nontrivial band crossings can be explored in angle-resolved
photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM).

## References

  * Bansil  _et al._ (2016) A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88, 021004 (2016).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Vergniory  _et al._ (2019) M. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, Nature 566, 480 (2019).
  * Hu  _et al._ (2019) J. Hu, S.-Y. Xu, N. Ni, and Z. Mao, Annual Review of Materials Research 49, 207 (2019).
  * Fu and Kane (2007) L. Fu and C. L. Kane, Physical Review B 76, 045302 (2007).
  * Singh  _et al._ (2012) B. Singh, A. Sharma, H. Lin, M. Z. Hasan, R. Prasad, and A. Bansil, Phys. Rev. B 86, 115208 (2012).
  * Wang  _et al._ (2013) Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Physical Review B 88, 125427 (2013).
  * Wang  _et al._ (2012) Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Physical Review B 85, 195320 (2012).
  * Lv  _et al._ (2015) B. Lv, N. Xu, H. Weng, J. Ma, P. Richard, X. Huang, L. Zhao, G. Chen, C. Matt, F. Bisti, _et al._ , Nature Physics 11, 724 (2015).
  * Xu  _et al._ (2013) S.-Y. Xu, C. Liu, S. Kushwaha, T.-R. Chang, J. Krizan, R. Sankar, C. Polley, J. Adell, T. Balasubramanian, K. Miyamoto, _et al._ , arXiv preprint arXiv:1312.7624 (2013).
  * Zhang  _et al._ (2019) C. Zhang, Y. Zhang, X. Yuan, S. Lu, J. Zhang, A. Narayan, Y. Liu, H. Zhang, Z. Ni, R. Liu, _et al._ , Nature 565, 331 (2019).
  * Zhang  _et al._ (2017) C. Zhang, E. Zhang, W. Wang, Y. Liu, Z.-G. Chen, S. Lu, S. Liang, J. Cao, X. Yuan, L. Tang, _et al._ , Nature communications 8, 1 (2017).
  * Kushwaha  _et al._ (2015) S. K. Kushwaha, J. W. Krizan, B. E. Feldman, A. Gyenis, M. T. Randeria, J. Xiong, S.-Y. Xu, N. Alidoust, I. Belopolski, T. Liang, _et al._ , APL materials 3, 041504 (2015).
  * Chiu and Schnyder (2015) C.-K. Chiu and A. P. Schnyder, in  _Journal of Physics: Conference Series_ , Vol. 603 (IOP Publishing, 2015) p. 012002.
  * Liang  _et al._ (2015) T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, and N. Ong, Nature materials 14, 280 (2015).
  * Button  _et al._ (1990) T. Button, I. McColm, and J. Ward, Journal of the Less Common Metals 159, 205 (1990).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Kokalj (1999) A. Kokalj, Journal of Molecular Graphics and Modelling 17, 176 (1999).
  * Blaha  _et al._ (2001) P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka, J. Luitz, _et al._ , An augmented plane wave+ local orbitals program for calculating crystal properties (2001).
  * Rourke and Julian (2012) P. Rourke and S. Julian, Computer Physics Communications 183, 324 (2012).
  * Parthe (1959) E. Parthe, Acta Crystallographica 12, 559 (1959).
  * Rodríguez-Carvajal (1993) J. Rodríguez-Carvajal, physica B 192, 55 (1993).
  * Matin  _et al._ (2018) M. Matin, R. Mondal, N. Barman, A. Thamizhavel, and S. Dhar, Physical Review B 97, 205130 (2018).
  * Mondal  _et al._ (2020) R. Mondal, S. Sasmal, R. Kulkarni, A. Maurya, A. Nakamura, D. Aoki, H. Harima, and A. Thamizhavel, Physical Review B 102, 115158 (2020).
  * He  _et al._ (2014) L. He, X. Hong, J. Dong, J. Pan, Z. Zhang, J. Zhang, and S. Li, Physical review letters 113, 246402 (2014).
  * Kumar  _et al._ (2017) N. Kumar, K. Manna, Y. Qi, S.-C. Wu, L. Wang, B. Yan, C. Felser, and C. Shekhar, Physical Review B 95, 121109 (2017).
  * Hu  _et al._ (2016) J. Hu, Z. Tang, J. Liu, X. Liu, Y. Zhu, D. Graf, K. Myhro, S. Tran, C. N. Lau, J. Wei, _et al._ , Physical review letters 117, 016602 (2016).
  * Hu  _et al._ (2017) J. Hu, Z. Tang, J. Liu, Y. Zhu, J. Wei, and Z. Mao, Physical Review B 96, 045127 (2017).
  * Du  _et al._ (2016) J. Du, H. Wang, Q. Chen, Q. Mao, R. Khan, B. Xu, Y. Zhou, Y. Zhang, J. Yang, B. Chen, _et al._ , Science China Physics, Mechanics & Astronomy 59, 657406 (2016).
  * Gao  _et al._ (2017) W. Gao, N. Hao, F.-W. Zheng, W. Ning, M. Wu, X. Zhu, G. Zheng, J. Zhang, J. Lu, H. Zhang, _et al._ , Physical review letters 118, 256601 (2017).
  * Butcher  _et al._ (2019) T. Butcher, J. Hornung, T. Förster, M. Uhlarz, J. Klotz, I. Sheikin, J. Wosnitza, and D. Kaczorowski, Physical Review B 99, 245112 (2019).
  * Wu  _et al._ (2019) F. Wu, C. Guo, M. Smidman, J. Zhang, Y. Chen, J. Singleton, and H. Yuan, npj Quantum Materials 4, 1 (2019).
  * Singha  _et al._ (2017) R. Singha, B. Satpati, and P. Mandal, Scientific reports 7, 1 (2017).
  * Inamdar  _et al._ (2013) M. Inamdar, M. Kriegisch, L. Shafeek, A. Sidorenko, H. Müller, A. Prokofiev, P. Blaha, and S. Paschen, in  _Solid State Phenomena_ , Vol. 194 (Trans Tech Publ, 2013) pp. 88–91.
  * Chen  _et al._ (2018) B. Chen, X. Duan, H. Wang, J. Du, Y. Zhou, C. Xu, Y. Zhang, L. Zhang, M. Wei, Z. Xia, _et al._ , npj Quantum Materials 3, 1 (2018).


# Probing the optical chiral response of single nanoparticles with optical
tweezers

R. Ali<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio de
Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brasil  R. S.
Dutra  LISComp-IFRJ, Instituto Federal de Educação, Ciência e Tecnologia, Rua
Sebastião de Lacerda, Paracambi, RJ,26600-000, Brasil  F. A. Pinheiro
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal
68528, Rio de Janeiro, RJ, 21941-972, Brasil  F. S. S. Rosa  Instituto de
Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de
Janeiro, RJ, 21941-972, Brasil  P. A. Maia Neto  Instituto de Física,
Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro,
RJ, 21941-972, Brasil

###### Abstract

We propose an enantioselective scheme to sort homogeneous chiral particles
using optical tweezers. For a certain range of material parameters, we show
that a highly focused circularly-polarized laser beam traps particles of a
specific chirality selected by the handedness of the trapping beam.
Furthermore, by applying a transverse Stokes drag force that displaces the
trapped particle off-axis, we allow for the rotation of the particle center-
of-mass around the trapping beam axis. The rotation angle is highly dependent
on the handedness of the trapped particle and is easily measurable with
standard video-microscopy techniques, allowing for an alternative mechanism
for chiral resolution. Our platform not only allows for enantioselection of
particles dispersed in solution but also paves the way to the characterization
of the chiral parameter of individual, homogeneous chiral microspheres using
optical tweezing.

##  I Introduction

Chiral structures are ubiquitous in nature and play a crucial role in the
biological functions of living species. Natural chiral substances have also
fundamental technological and industrial importance as they are present in
many chemical and pharmaceutical compounds wagniere . Natural optical activity
is the hallmark optical effect exhibited by chiral media, as discovered in the
pioneering work by Arago, and it is typically weak for naturally occurring
materials wagniere . With advent of metamaterials and plasmonics, new
artificial media with much higher chiral optical response have been developed
to generate novel optical properties and applications such as negative index
media pendry2004 ; plum , broadband circular polarizers gansel2009 , and
enantioselective photochemistry tang2011 . An important class of artificial
chiral media is composed of plasmonic nanoparticles, which includes chiral
nanocrystals, DNA-assembled plasmonic nanostructures, chiral plasmonic
particles assembled on scaffolds, core-shell plasmonic spheres cpreview1 ;
cpreview2 ; cpreview3 , and assemblies of nanoparticles in chiral
configurations, such as helices fan2010  or even random arrangements
pinheiro2017 . However, it is quite challenging to probe individual chiral
optical response for single chiral plasmonic nanoparticles, as the existing
enantioselective methods usually can only measure the average chiral optical
response of a large number of (typically) non-identical particles in solution
(e.g. circular dichroism). By the same token, no established method exists to
determine the chiral parameter $\kappa$ of individual nanoparticles, which
would be of great interest to characterise optical chiral response in chiral
nanoplasmonics dionnereply ; Mastroianni .

To circumvent these limitations, we have recently put forward an alternative
chiral resolution method to not only achieve enantioselection of chiral
nanoparticles but also to determine their chiral parameter $\kappa$ rali2020 .
This method is based on optical tweezing of dielectric spheres coated with
chiral shells, where it is possible to select the handedness of the trapped
coated particles by choosing the appropriate circular polarization of the
trapping laser beam. Our approach compares favorably with previous proposals
li2007 ; spivak2009 ; li2010 ; cameron2014 ; durand2013 ; bradshaw2014 ;
wang2014 ; hayat2015 ; alizadeh2015 ; durand2016 ; chen2016 ; dionne2016 ;
zhang2017 ; acebal2017 ; ho2017 ; cao2018 ; dionne2018  and implementations
hernandez2013 ; tkachenko2014 ; donato2015 ; tkachenko2014b ; dionne2017 ;
Schnoering2018 ; krevets2019 ; Nker2019  of optical chiral resolution as it
allows for the determination of the chiral parameter of each individual
trapped nanoparticle via the rotation of the equilibrium position under the
effect of a transverse drag force. However, this method has only been applied
so far to coated nanoparticles. The case of homogeneous chiral nanoparticles
is not only different from a physical point of view, as their light scattering
properties are distinct from coated ones, but also is more relevant for a
number of chiral systems, such as chiral plasmonic nanocrystals fan2012 . Here
we show that our method can be applied to micrometer sized homogeneous chiral
spheres. In addition, we provide a detailed derivation of our theoretical
approach, which is based on a generalization of the Mie-Debye theory of
optical tweezers epl ; Mazolli2003  to the case of chiral trapped particles.

This paper is organized as follows. In section II, we review the basic
formalism of electromagnetic scattering by a chiral sphere using the formalism
of Debye potentials. In addition, the optical torque exerted on trapped chiral
particles is discussed in detail. In Sec. III, we present our results and
discuss how the chirality affects the condition for optical tweezing. Finally,
in Sec. IV we summarise our findings and conclusions.

##  II Methodology

###  II.1 Electromagnetic fields in chiral media

In a chiral medium, the electromagnetic fields ${\bf E}$ and ${\bf H}$ are
coupled by a chirality parameter $\kappa$. The constitutive relations may be
written as lindell ; lakhtakia ; chan2014

$\displaystyle\textbf{D}=\epsilon_{0}\epsilon\textbf{E}+i\kappa\sqrt{\epsilon_{0}\mu_{0}}\,{\bf
H}$
$\displaystyle\textbf{B}=-i\kappa\sqrt{\epsilon_{0}\mu_{0}}\,\textbf{E}+\mu\mu_{0}\textbf{H}$
(1)

where ${\bf D}$ and ${\bf B}$ are the electric displacement and the magnetic
field in the chiral material, respectively. Furthermore, $\epsilon$, $\mu$ are
the effective relative permittivity and permeability of the medium, and
$\epsilon_{0}$, $\mu_{0}$ are the absolute permittivity and permeability of
vacuum.

The Maxwell’s equations for chiral media in the frequency domain can be
compactly written in a matrix form with the help of the constitutive relations
(1) as

$\displaystyle\nabla\cdot\begin{bmatrix}\textbf{E}\\\
{\textbf{H}}\end{bmatrix}=0\hskip 15.0pt,\hskip
15.0pt\nabla\\!\times\\!\begin{bmatrix}\textbf{E}\\\
{\textbf{H}}\end{bmatrix}=K\\!\begin{bmatrix}\textbf{E}\\\
{\textbf{H}}\end{bmatrix}$ (2)

where

$K=\begin{bmatrix}k_{0}\kappa&ik_{0}\mu{\sqrt{\mu_{0}/\epsilon_{0}}}\\\
{-ik_{0}\epsilon\sqrt{\epsilon_{0}/\mu_{0}}}&k_{0}\kappa\end{bmatrix}.$

and $k_{0}=\omega\sqrt{\mu_{0}\epsilon_{0}}=\omega/c$. In addition, a direct
inspection of (1) shows that the transverse character of E and H is only
ensured if $\mu\epsilon\neq\kappa^{2}$, so we assume that such condition holds
from now onwards footnote . By introducing a linear transformation $A$, also
known as Bohren decomposition, that mixes E and H bohren-huffman ; lakhtakia

$\displaystyle\begin{bmatrix}\bf{\Lambda_{+}}\\\
\bf{\Lambda_{-}}\end{bmatrix}{\small=\frac{1}{2}\begin{bmatrix}\sqrt{\mu_{0}/\epsilon_{0}}&i\sqrt{\mu/\epsilon}\\\
i\sqrt{\epsilon/\mu}&\sqrt{\epsilon_{0}/\mu_{0}}\end{bmatrix}}\begin{bmatrix}\bf{E}\\\
{\bf{H}}\end{bmatrix}=A\begin{bmatrix}\bf{E}\\\ {\bf{H}}\end{bmatrix}\,,$ (3)

it is simple to show that we get uncoupled Helmholtz equations for each of the
vectors ${\bf\Lambda}_{\sigma},\;\sigma=\pm 1:$

$\displaystyle\nabla^{2}{\bf{\Lambda}_{\sigma}}+k_{\sigma}^{2}\bf{\Lambda}_{\sigma}=0$
(4)

where $k_{\sigma}=k_{0}(\sqrt{\mu\epsilon}-\sigma\kappa).$ The parameter
$\sigma$ represents the helicity of the propagating waves, as
$\bf{\Lambda}_{+}$ and $\bf{\Lambda}_{-}$ are left- and right-hand eigenmodes
of incident circularly polarized waves that independently satisfy the
Maxwell’s equations in the frequency domain. Finally, given the dispersion
relation for $k_{\sigma}$, it is convenient to define a chiral refraction
index $m_{\sigma}$

$m_{\sigma}=\sqrt{\mu\epsilon}-\sigma\kappa$ (5)

that effectively characterizes the propagation of circularly polarized waves
of helicity $\sigma.$

Figure 1:  (a) Schematic representation of enantioselective optical tweezing.
A single circularly-polarized Gaussian laser beam is focused at the focal
plane of the objective lens in order to optically trap homogeneous chiral
microspheres dispersed in aqueous solution. The sample is driven laterally at
a constant speed. (b) As the trapped sphere is laterally displaced by the
resulting Stoles drag force $F_{S}$, its equilibrium position rotates around
the optical $z$ axis by an angle $\alpha.$ The cylindrical optical force
components $F_{\rho}$ and $F_{\phi}$ are shown.

###  II.2 Mie series solution for a chiral sphere

The solution of the Helmholtz equation (4) in spherical coordinates is
thoroughly discussed in the literature bohren-huffman ; Born-Wolf , but the
chiral case presents a non-trivial additional element: while propagating
eigenmodes correspond to the vector fields ${\bf\Lambda}_{\sigma}$, the
boundary conditions involve the fields ${\bf E,D,B}$ and ${\bf H}$.
Fortunately, the linear relation (3) allows a straightforward solution for the
scattered field by a chiral sphere (given an incident plane wave) bohren-
huffman ; lakhtakia ; bohren :

$\displaystyle{\bf
E}_{s}=E_{0}\sum^{\infty}_{\ell}(i)^{{\ell}}\frac{2{\ell}+1}{{\ell}({\ell}+1)}(ia_{\ell}N_{e1{\ell}}^{3}$
(6) $\displaystyle-
b_{\ell}M_{o1{\ell}}^{3}+c_{\ell}M_{e1{\ell}}^{3}-id_{\ell}N_{o1{\ell}}^{3})$
$\displaystyle{\bf
H}_{s}=\frac{k}{\omega\mu}E_{0}\sum^{\infty}_{\ell}(i)^{{\ell}}\frac{2{\ell}+1}{{\ell}({\ell}+1)}(a_{\ell}M_{e1{\ell}}^{3}$
$\displaystyle+ib_{\ell}N_{o1{\ell}}^{3}-ic_{\ell}N_{e1{\ell}}^{3}-d_{\ell}M_{o1{\ell}}^{3}),$

where $M_{e1\ell}^{3}$, $M_{o1\ell}^{3}$, $N_{e1\ell}^{3}$ and
$N_{o1\ell}^{3}$ are the vector spherical harmonics bohren-huffman . The
scattering amplitudes $a_{\ell}$, $b_{\ell}$, $c_{\ell}$ and $d_{\ell}$ are
commonly known as Mie coefficients bohren-huffman ; lakhtakia ; bohren ; shang
; chan2014

$\displaystyle
a_{\ell}=\frac{V_{\ell}(-)A_{\ell}(+)+V_{\ell}(+)A_{\ell}(-)}{W_{\ell}(+)V_{\ell}(-)+V_{\ell}(+)W_{\ell}(-)}$
$\displaystyle
b_{\ell}=\frac{W_{\ell}(+)B_{\ell}(-)+W_{\ell}(-)B_{\ell}(+)}{W_{\ell}(+)V_{\ell}(-)+V_{\ell}(+)W_{\ell}(-)}$
$\displaystyle
c_{\ell}=-d_{\ell}=i\frac{W_{\ell}(-)A_{\ell}(+)-W_{\ell}(+)A_{\ell}(-)}{W_{\ell}(+)V_{\ell}(-)+V_{\ell}(+)W_{\ell}(-)}$
(7)

with

$\displaystyle
W_{\ell}(\sigma)=m\psi_{\ell}(y_{\sigma})\xi_{\ell}^{{}^{\prime}}(x)-\xi_{\ell}(x)\psi_{\ell}^{{}^{\prime}}(y_{\sigma})$
(8) $\displaystyle
V_{\ell}(\sigma)=\psi_{\ell}(y_{\sigma})\xi_{\ell}^{{}^{\prime}}(x)-m\xi_{\ell}(x)\psi_{\ell}^{{}^{\prime}}(y_{\sigma})$
$\displaystyle
A_{\ell}(\sigma)=m\psi_{\ell}(y_{\sigma})\psi_{\ell}^{{}^{\prime}}(x)-\psi_{\ell}(x)\psi_{\ell}^{{}^{\prime}}(y_{\sigma})$
$\displaystyle
B_{\ell}(\sigma)=\psi_{\ell}(y_{\sigma})\psi_{\ell}^{{}^{\prime}}(x)-m\psi_{\ell}(x)\psi_{\ell}^{{}^{\prime}}(y_{\sigma})$

where $m=m_{+}m_{-}/2(m_{+}+m_{-}).$ The Riccati-Bessel functions Abramowitz
$\psi_{\ell}$, $\xi_{\ell}$ are evaluated either at the size parameter
$x=\sqrt{\epsilon_{w}}k_{0}a$ defined with respect to the wavelength in the
non-magnetic achiral host medium (relative electric permittivity
$\epsilon_{w}$) or at $y_{\sigma}=m_{\sigma}x/\sqrt{\epsilon_{w}}.$ Finally,
for future convenience, we introduce the Debye potentials $\Pi^{E}$ and
$\Pi^{M}$ for electric and magnetic multipoles Born-Wolf

$\displaystyle\Pi^{E}({\bf r})=\sum^{\infty}_{\ell}\frac{({\bf r}\cdot{\bf
E})}{{\ell}({\ell}+1)}$ (9) $\displaystyle\Pi^{M}({\bf
r})=\sum^{\infty}_{\ell}\frac{({\bf r}\cdot{\bf H})}{{\ell}({\ell}+1)}\,,$

which satisfy a (scalar) Helmholtz equation. Using Eq. (9) we write the
electromagnetic fields in terms of the scalar Debye potentials as follows:

$\displaystyle{\bf E}=\nabla\times\nabla\times\left({\bf
r}\,\Pi^{E}\right)+i\omega\mu\,\nabla\times\left({\bf r}\,\Pi^{M}\right)$ (10)
$\displaystyle{\bf H}=\nabla\times\nabla\times\left({\bf
r}\,\Pi^{M}\right)-i\omega\epsilon\,\nabla\times\left({\bf
r}\,\Pi^{E}\right)\,.$

###  II.3 Mie-Debye theory of optical tweezers for chiral sphere

As it is depicted in Fig. 1, an optical tweezers setup consists of a laser
beam strongly focused by a high numerical aperture (NA) objective. Stable
trapping in the focal region is possible for certain “goldilocks” combinations
of the optical properties of the object to be trapped and the surrounding
medium. We consider a $1064\,\rm nm$ Gaussian beam, circularly polarized with
helicity $\sigma$, at the objective entrance port. The beam is focused by the
objective into the sample region containing an aqueous suspension (refractive
index $n_{w}=\sqrt{\epsilon_{w}}$) of homogeneous chiral spheres, with
effective relative permittivity $\epsilon_{s}$ and chirality $\kappa$. The
focused beam in the sample region is represented in terms of the so called
Richards-Wolf integral representation RichardsWolf ; Mazolli2003

$\displaystyle\mathbf{E}_{\rm in}(\mathbf{r})$
$\displaystyle={E}_{0}\int_{0}^{2\pi}d\varphi_{k}\int_{0}^{\theta_{0}}d{\theta_{k}}\sin\theta_{k}\sqrt{\cos\theta_{k}}\,e^{-\gamma_{f}^{2}\sin^{2}\theta_{k}}$
(11) $\displaystyle\hskip
60.0pt\times\,e^{i\mathbf{k}(\theta_{k},\varphi_{k})\cdot(\mathbf{r}+\mathbf{r}_{s})}\mathbf{\hat{\epsilon}_{\sigma}^{\prime}}(\theta_{k},\varphi_{k}),$

where we set the origin at the center of the sphere and the focal point is at
position $-\mathbf{r}_{s}$. The region of integration in Fourier space is
defined by the angle $\theta_{0},$ which in turn depends on the objective
numerical aperture NA: $\sin\theta_{0}=\min\\{1,\mbox{NA}/n_{w}\\}.$
$\gamma_{f}$ is the ratio of the objective focal length to the laser beam
waist at the objective entrance port and
$\mathbf{\hat{\epsilon}_{\sigma}^{\prime}}(\theta_{k},\varphi_{k})=\mathbf{\hat{x}^{\prime}}(\theta_{k},\varphi_{k})+i\sigma\mathbf{\hat{y}^{\prime}}(\theta_{k},\varphi_{k}).$
The unit vectors $\mathbf{\hat{x}^{\prime}}(\theta_{k},\varphi_{k})$ and
$\mathbf{\hat{y}^{\prime}}(\theta_{k},\varphi_{k})$ are obtained from
$\mathbf{\hat{x}}$ and $\mathbf{\hat{y}}$ by rotation with Euler angles
$\alpha=\varphi_{k},$ $\beta=\theta_{k}$ and $\gamma=-\varphi_{k}$. We have
assumed an ideal aplanatic optical system when writing Eq. (11). Optical
aberrations can be included as additional phase factors in the integrand in
the r.-h.-s. of (11) Dutra2014  and the resulting optical force can be derived
by following the steps presented below.

We take $\mathbf{E}_{\rm in}(\mathbf{r})$ as the incident field on the chiral
microsphere. The strategy is to take each plane wave component of Eq. (11),
solve the scattering problem using the results of section II.2, and superpose
the solutions appropriately. There is, however, a caveat: the expressions in
last section assume a $\hat{\bf z}$-propagating incident wave, and in (11) we
have components propagating in many different directions, so we must rotate
our coordinate system in order to have the solution for an arbitrary incident
wave. This is accomplished by the matrix elements of finite rotations
$d^{j}_{m,m^{\prime}}(\theta)$Edmonds . After these operations, the Debye
potentials associated to an incident circular polarized plane wave (helicity
$\sigma$) propagating along an arbitrary direction defined by the spherical
angles $(\theta_{k},\phi_{k})$ are given by

$\displaystyle\Pi^{E,{\rm in}}_{{\bf
k},\sigma}(r,\theta,\phi)=\sigma\frac{iE_{0}}{k}\sum_{{\ell}}i^{{\ell}}j_{\ell}{(kr})\sqrt{\frac{4\pi(2{\ell}+1)}{{\ell}({\ell}+1)}}Y^{(\sigma)}_{\ell}(\theta,\phi)$
(12) $\displaystyle\Pi^{M,{\rm in}}_{{\bf
k},\sigma}(r,\theta,\phi)=\frac{E_{0}}{k}\sum_{{\ell}}i^{{\ell}}j_{\ell}{(kr})\sqrt{\frac{4\pi(2{\ell}+1)}{{\ell}({\ell}+1)}}Y^{(\sigma)}_{\ell}(\theta,\phi)\,,$

where $j_{\ell}(x)$ are the spherical Bessel functions Abramowitz  and

$Y^{(\sigma)}_{\ell}(\theta,\phi):=\sum^{\ell}_{{m}=-\ell}Y_{\ell{m}}(\theta,\phi)e^{-i\phi_{k}(m-\sigma)}d^{\ell}_{m,{\sigma}}(\theta_{k})\,.$

Solving the scattering problem for general $\theta_{k}$ and $\phi_{k}$,
superposing the solutions according to (11), and undertaking a translation to
a coordinate system centered at the focal point — so that the sphere’s center
is finally at position $\mathbf{r}_{s}$ —, we arrive at the Debye potentials
for the total fields

$\displaystyle\Pi_{\sigma}^{E,\rm{tot}}=\sigma\frac{iE_{0}}{k}\sum_{\ell,m}i^{\ell}\gamma_{\ell,m}^{\sigma}\left[j_{\ell}{(kr})-(a_{\ell}+i\sigma
d_{\ell})h_{\ell}(kr)\right]\times$ $\displaystyle\hskip 60.0ptY_{\ell
m}(\theta,\phi)e^{-i(m-\sigma)\phi_{s}}$ (13)
$\displaystyle\Pi_{\sigma}^{M,\rm{tot}}=\frac{E_{0}}{k}\sum_{\ell,m}i^{\ell}\gamma_{\ell,m}^{\sigma}\left[j_{\ell}{(kr})-(b_{\ell}-i\sigma
c_{\ell})h_{\ell}(kr)\right]\times$ $\displaystyle\hskip 60.0ptY_{\ell
m}(\theta,\phi)e^{-i(m-\sigma)\phi_{s}}\,,$ (14)

where $h_{\ell}(kr)$ is the spherical Hankel function of the first kind. The
coefficients

$\displaystyle\gamma_{\ell,m}^{\sigma}=2\pi\sqrt{\frac{4\pi(2{\ell}+1)}{{\ell}({\ell}+1)}}(-i)^{m-\sigma}\int_{0}^{\theta_{0}}d\theta_{k}\sin\theta_{k}\times$
$\displaystyle\sqrt{\cos\theta_{k}}d^{\ell}_{m,\sigma}(\theta_{k})J_{m-\sigma}(k\rho_{s}\sin\theta_{k})e^{ikz_{s}\cos{\theta_{k}}}$
(15)

are written in terms of the cylindrical Bessel functions $J_{m-\sigma}$ of
integer order $m-\sigma.$

Here we have expressed the microsphere position ${\bf
r}_{s}=(\rho_{s},\varphi_{s},z_{s})$ in cylindrical coordinates. When taking
$\kappa$ to zero ($c_{\ell}=d_{\ell}=0$) we recover the Debye potentials for
an achiral sphere Mazolli2003 . However, the most interesting property of Eqs.
(14) is that they have the same form as the Debye potentials of an achiral
sphere, provided the following replacements are made

$\displaystyle A_{\ell}\leftrightarrow a_{\ell}+\,i\,\sigma d_{\ell}$ (16)
$\displaystyle B_{\ell}\leftrightarrow b_{\ell}-i\sigma\,c_{\ell}\,,$ (17)

where $A_{\ell}$ and $B_{\ell}$ would be the Mie coefficients of the
equivalent achiral sphere scattering a circularly polarized beam of helicity
$\sigma.$ Equations (16) and (17) show that one can map the problem of light
scattering by a homogeneous chiral sphere into the simpler original Mie
scattering problem.

As the final step in the derivation, we compute the optical force upon the
sphere by integration of the Maxwell stress tensor:

${\bf F}=\lim_{r\rightarrow\infty}\left[-\frac{r}{2}\int_{S}{\bf
r}\left(\epsilon_{w}\epsilon_{0}E_{\rm tot}^{2}+\mu_{0}H_{\rm
tot}^{2}\right)d\Omega\right]$ (18)

where ${\bf E}_{\rm tot}$ and ${\bf B}_{\rm tot}$ are the total electric and
magnetic fields and the Gaussian spherical surface $S$ is taken at infinity.
At this point it is useful to define the (dimensionless) efficiency factor
${\bf Q}$

${\bf Q}=\frac{c}{n_{w}P}{\bf F}\,,$ (19)

where ${\bf F}$ is given by (18), $P$ is the incident laser power at the
sample region and $c$ is speed of light. The efficiency may be conveniently
split into extinction and scattering terms

${\bf Q}={\bf Q}_{e}+{\bf Q}_{s},$ (20)

where the extinction term ${\bf Q}_{e}$ is connected to the rate of momentum
removal from the incident beam. Part of this momentum ends up in the scattered
field at a (normalized) rate $-{\bf Q}_{s}$, so the total (normalized)
momentum transfer rate to the particle is ${\bf Q}_{e}+{\bf Q}_{s}$. After a
long but straightforward calculation epl ; Mazolli2003 , we get the
expressions for the efficiency components $Q_{z}$, $Q_{s}$ and $Q_{\phi}$,
related to the axial, radial and azimuthal force components, respectively.
Explicitly, we have

  * •

Extinction components

$Q_{ez}(\rho_{s},z_{s})=\frac{4\gamma_{f}^{2}}{AN}{\rm Re}\sum_{\ell
m}(2\ell+1)G^{(\sigma)}_{\ell,m}(A_{\ell}+B_{\ell})G^{\prime(\sigma)*}_{\ell,m},$
(21)

$Q_{e\rho}(\rho_{s},z_{s})=\frac{2\gamma_{f}^{2}}{AN}{\rm Im}\sum_{\ell
m}(2\ell+1)G^{(\sigma)}_{\ell,m}(A_{\ell}+B_{\ell})\left(G^{(\sigma)-}_{\ell,m+1}-G^{(\sigma)+}_{\ell,m-1}\right)^{*}$
(22)

$Q_{e\phi}(\rho_{s},z_{s})=-\frac{2\gamma_{f}^{2}}{AN}{\rm Re}\sum_{\ell
m}(2\ell+1)G^{(\sigma)}_{\ell,m}(A_{\ell}+B_{\ell})\left(G^{(\sigma)+}_{\ell,m-1}+G^{(\sigma)-}_{\ell,m+1}\right)^{*}$
(23)

  * •

Scattering components

$\displaystyle Q_{sz}(\rho_{s},z_{s})$
$\displaystyle=-\frac{8\gamma_{f}^{2}}{AN}{\rm Re}\sum_{\ell
m}\frac{\sqrt{\ell(\ell+2)(\ell+m+1)(\ell-m+1)}}{\ell+1}(A_{\ell}A_{\ell+1}^{*}+B_{\ell}B_{\ell+1}^{*})G^{(\sigma)}_{\ell,m}G^{(\sigma)*}_{\ell+1,m}$
(24) $\displaystyle-\frac{8\gamma_{f}^{2}}{AN}\,\sigma\,{\rm Re}\sum_{\ell
m}\frac{(2\ell+1)}{\ell(\ell+1)}mA_{\ell}B_{\ell}^{*}|G^{(\sigma)}_{\ell,m}|^{2},$

$\displaystyle Q_{s\rho}(\rho_{s},z_{s})=\frac{4\gamma_{f}^{2}}{AN}\sum_{\ell
m}\frac{\sqrt{\ell(\ell+2)(\ell+m+1)(\ell+m+2)}}{\ell+1}{\rm
Im}\biggl{\\{}(A_{\ell}A_{\ell+1}^{*}+B_{\ell}B_{\ell+1}^{*})\left[G^{(\sigma)}_{\ell,m}G^{(\sigma)*}_{\ell+1,m+1}\right.$
$\displaystyle\left.+G^{(\sigma)}_{\ell,-m}G^{(\sigma)*}_{\ell+1,-(m+1)}\right]\biggl{\\}}-\frac{8\gamma_{f}^{2}}{AN}\sigma\sum_{\ell
m}\frac{(2\ell+1)}{\ell(\ell+1)}\sqrt{(\ell-m)(\ell+m+1)}\;{\rm
Re}(A_{\ell}B_{\ell}^{*})\,{\rm
Im}(G^{(\sigma)}_{\ell,m}G^{(\sigma)*}_{\ell,m+1})$ (25)

$\displaystyle Q_{s\phi}(\rho_{s},z_{s})=-\frac{4\gamma_{f}^{2}}{AN}\sum_{\ell
m}\frac{\sqrt{\ell(\ell+2)(\ell+m+1)(\ell+m+2)}}{\ell+1}{\rm
Re}\biggl{\\{}(A_{\ell}A_{\ell+1}^{*}+B_{\ell}B_{\ell+1}^{*})\left[G^{(\sigma)}_{\ell,m}G^{(\sigma)*}_{\ell+1,m+1}\right.$
$\displaystyle\left.-G^{(\sigma)}_{\ell,-m}G^{(\sigma)*}_{\ell+1,-(m+1)}\right]\biggr{\\}}+\frac{8\gamma_{f}^{2}}{AN}\sigma\sum_{\ell
m}\frac{(2\ell+1)}{\ell(\ell+1)}\sqrt{(\ell-m)(\ell+m+1)}\;{\rm
Re}(A_{\ell}B_{\ell}^{*})\,{\rm
Re}(G^{(\sigma)}_{\ell,m}G^{(\sigma)*}_{\ell,m+1})\,,$ (26)

We have defined $A=1-e^{-2\gamma_{f}^{2}\,\sin^{2}\theta_{0}}$ as the fraction
of the trapping beam power that fills the objective entrance aperture and
propagates into the sample. Explicit expressions for the multipole
coefficients $G_{\ell m}^{(\sigma)}$, ${G^{\prime}}_{\ell m}^{(\sigma)}$ and
$G_{\ell m}^{(\sigma)\pm}$ are given in the appendix.

##  III Results and Discussion

In this section, we present our numerical results for the optical forces and
optical torques in the framework of the MDSA formalism derived in Sec. II. In
all the following examples we take the refractive indexes of the sphere and
aqueous suspension to be $n_{s}=1.58$ and $n_{w}=1.332,$ respectively. The
vacuum wavelength of the laser beam is $\lambda_{0}=2\pi/k_{0}=1064\,\rm nm$,
and the other relevant parameters are $\gamma_{f}=1.226$ and NA$=1.4$.

###  III.1 Optical trapping

Due to axial symmetry, it is sufficient to analyze the optical force $F_{z}$
to ensure 3-D trapping stability of optical trap. In Fig. 2, we calculate the
normalized axial force acting on a microsphere as a function of sphere
position along the laser axis when taking right-circular polarization (RCP),
which corresponds to helicity $\sigma=-1.$. We clearly see an enantioselection
effect taking place. For instance, the RCP beam attracts the right-handed
chiral particles with $\kappa=-0.3$ (black line) towards the paraxial focal
plane with stable equilibrium position close to the focus and repels the left-
handed chiral enantiomer with $\kappa=0.3$ out of the illuminated area (dashed
line).

Figure 2:  Dimensionless axial force efficiency $Q_{z}$ as a function of
position (in units of sphere radius) along the laser axis for different values
of the chirality parameter: $\kappa=-0.3$ (black line), $\kappa=0$ (dotted
line) and $\kappa=0.3$ (dashed line). The incident beam is right-circularly
polarized ($\sigma=-1$) and the microsphere radius is $a=1\,\mu{\rm m}$.
Inset: $Q_{z}$ as a function of $\kappa$ at a fixed axial position $z/a=1.2$
for $\sigma=-1$ (red dotted line) and $\sigma=1$ (blue line).

Due to mirror symmetry, this result is exactly reversed for left circularly
polarized (LCP) beams, so that in this case the spheres with positive
chirality get trapped. This is illustrated by the inset of Fig. 2, where we
plot the normalized axial force as a function of chirality for left and right
polarization of the beam. Therefore, in a racemic mixture one could
potentially use this effect to trap exclusively right- or left-handed
particles by using a RCP or a LCP beam. In addition, from the inset of Fig. 2
it is also clear that despite favoring “same sign” trapping (RCP traps right-
handed particles and vice versa), “oposite sign” trapping is also possible up
to a critical value $|\kappa_{0}|\approx 0.22$.

We analyse more thoroughly the interplay of chirality and particle size in
Fig. 3. The colored area shows the negative optical force (trapping force) on
right handed chiral enantiomers for RCP incident light. The most important
feature of this plot is the weak dependence of the force upon the sphere
radius, at least for the range considered. This means that, at least for
sufficiently large values of $\kappa$, the enantioselection effect is quite
robust against variation of the radius, and could be effective even in
solutions where the spheres size dispersion is large. It is important to
emphasize that such values of $\kappa$ are within the reach of state-of-the-
art plasmonic and metamaterial chiral nanoparticles dionne2016 ; fan2012 ;
wang2014 .

Figure 3:  Density plot of the dimensionless axial force efficiency $Q_{z}$ at
axial position $z/a=1.2$. The horizontal and vertical axes represent the
chirality parameter and the microsphere radius, respectively. Only negative
values of $Q_{z}$ are represented by the color scale. The vertical line
indicates the critical value $\kappa_{0}$ above which no trapping takes place
as the optical force becomes positive.

###  III.2 Optical torque

In the preceding section, we have shown that one can perform chiral resolution
of a racemic mixture by selective trapping of chiral enantiomers that match
the handedness of the laser beam. In this section, we discuss another method
that is based on the transfer of spin angular momentum (SAM) from the focused
beam to the trapped microsphere, allowing to determine the chiral parameter of
single particles.

In order to exploit the SAM transfer, first we stably trap a microsphere and
then produce a water flow by driving the sample laterally (say, along the
$x-$direction), as indicated in Fig. 1. A drag force $F_{S}$ is then generated
along the $x-$direction, that displaces the sphere away from the center of the
optical trap and gives rise to a restoring force $F_{\rho}$. In addition, when
the trapping beam is circularly polarized, it is intuitive that by pushing the
sphere away from the beam axis we should also get a force along the azimuthal
direction $F_{\phi}$, as illustrated in Fig. 1. This azimuthal force component
essentially converts (some of) the spin angular momentum of the beam to
angular momentum of the sphere, until it equilibrates at an angle
$\alpha=\arctan(F_{\phi}/F_{\rho})$. Explicit expressions for such force
components were given in Sec. II. It is more than reasonable to expect that
such equilibrium angle should depend considerably upon the sphere chirality,
and that is precisely what we show below.

Figure 4: Rotation angle $\alpha$ of a chiral microsphere of radius
$a=0.6\,\mu{\rm m}$ as a function of the chirality parameter $\kappa.$ The
incident beam is right-circularly polarized ($\sigma=-1$).

Fig. 4 shows the optical torque exerted on the sphere in terms of the rotation
angle $\alpha$ in degrees for a RCP trapping beam ($\sigma=-1$). First of all,
it is important to mention that for a small decrease in the chirality
parameter of magnitude $\delta\kappa=10^{-2}$ the sphere undergoes a
measurable additional rotation of $\delta\alpha=0.2-0.5$ degrees (with the
largest variation for the highest positive values of $\kappa$ shown in the
figure). Such strong variation of the rotation angle suggests a suitable
scheme to determine and characterize the chiral parameter of a single
particle, which remains a challenging task so far dionnereply .

When the rotation angle becomes positive for negative values of $\kappa$ in
Fig. 4, it means that the scattered field carries a magnitude of angular
momentum that exceeds that of the incident field kaina2019 , so that
conservation of angular momentum leads to a torque opposite to the input spin.
Such negative torque corresponds to a positive rotation angle $\alpha$ because
we take $\sigma=-1$ here. By increasing the value of $\kappa$, the rotation
angle not only changes its sign but also achieves very large values in
comparison to the case of achiral homogenous spheres treated so far kaina2019
, showing that chirality leads to a significant enhancement of the light-
particle angular momentum transfer.

Figure 4 also shows that $\alpha$ changes its sign at a relative small value,
$\kappa\approx-0.02$. This result demonstrates that enantioselection can also
be attained by detecting the sign of the rotation angle of a particle trapped
using optical tweezers, in addition to the previous chiral resolution
mechanism based on selective trapping of particles of a given handedness, as
previously presented.

Figure 5: Microsphere rotation angle versus sphere radius. The solid and
dotted lines correspond to $\kappa=-0.3$ and $\kappa=0,$ respectively. The
incident beam is right-circularly polarized ($\sigma=-1$).

In Fig. 5, we plot the variation of the rotation angle $\alpha$ versus the
microsphere radius for two fixed values of the chirality parameter:
$\kappa=-0.3$ (solid) and $\kappa=0$ (dotted). As in Fig. 4, we take RCP
corresponding to $\sigma=-1.$ For small microspheres the rotation angle is
negative for achiral particles (positive torque) and positive for
$\kappa=-0.3$ (negative torque). In the Rayleigh scattering regime,
$a\ll\lambda_{0}$, scattering is weak and both force and torque are dominated
by the extinction contribution. Indeed, very small particles behave as local
probes of the input angular momentum, thus explaining the negative angles in
the case of small achiral spheres. On the other hand, Fig. 5 shows that the
particle chirality strongly affects the sign of the optical torque even in the
Rayleigh limit.

As the size parameter increases light scattering becomes more prominent and
the transfer of angular momentum is amplified SBradshaw2017 , leading to much
larger angles as illustrated by Fig. 5. In particular, we find a peak value
around $a\approx 0.3\,\mu{\rm m}$ of approximately $17^{\rm o}.$ Such a large
rotation angle cannot be obtained with achiral microspheres kaina2019 . As the
radius is further increased into the ray optics range, the rotation angle for
the achiral sphere oscillates around zero, which is compatible with the
vanishing optical torque in the ray optics approximation Mazolli2003 , whereas
the curve for the chiral particle suggests a nonzero torque in this limit.

##  IV Conclusions

In this paper we developed the theory of optical tweezing for optically active
homogeneous microspheres. First we developed the Bohren decomposition in terms
of Debye potentials representing the incident and scattered field. Then we
used the Maxwell stress tensor to calculate the optical forces exerted on
chiral spheres. Our numerical analysis predicts that circularly polarized
light exerts different optical forces on left- and right-handed chiral
microspheres, allowing for enantioselection. We also have shown that angular
momentum transfer is different for chiral microspheres of opposite handedness,
which not only greatly enhances optical rotation of the microsphere with
respect to the achiral case treated so far but also leads to another chiral
resolution method based on enantioselective optical torque. Altogether our
findings pave the way for alternative chiral resolution methods based on
optical tweezing and characterisation of chirality for the important class of
homogeneous chiral microspheres, each and every one with its individual,
unique chiral response.

###### Acknowledgements.

We thank D. S. Ether jr and N. B. Viana for inspiring discussions. This work
has been supported by the Brazilian agencies National Council for Scientific
and Technological Development (CNPq), Coordination for the Improvement of
Higher Education Personnel (CAPES), the National Institute of Science and
Technology Complex Fluids (INCT-FCx), and the Research Foundations of the
States of Minas Gerais (FAPEMIG), Rio de Janeiro (FAPERJ) and São Paulo
(FAPESP).

##  Appendix A  Multipole coefficients of the focused beam

The multipole coefficients in expressions (21)-(26) are given by

$G_{\ell
m}^{(\sigma)}({\rho_{s},z_{s}})=\int_{0}^{\theta_{0}}d\theta\,\sin\theta\sqrt{\cos{\theta}}\,e^{-\gamma_{f}^{2}\sin^{2}\theta}d_{m,\sigma}^{\ell}(\theta)\,J_{m-\sigma}\left(k\rho_{s}\sin\theta\right)e^{i\,k\cos\theta\,z_{s}},$
(27)

$G^{\prime(\sigma)}_{\ell,m}({\rho_{s},z_{s}})=\int_{0}^{\theta_{0}}d\theta\,\sin\theta(\cos\theta)^{3/2}\,e^{-\gamma_{f}^{2}\sin^{2}\theta}d_{m,\sigma}^{\ell}(\theta)\,J_{m-\sigma}\left(k\rho_{s}\sin\theta\right)e^{i\,k\cos\theta\,z_{s}},$
(28)

$G_{\ell,m}^{(\sigma)\pm}({\rho_{s},z_{s}})=\int_{0}^{\theta_{0}}{d\theta}\,\sin^{2}\theta\sqrt{\cos\theta}\,e^{-\gamma_{f}^{2}\sin^{2}\theta}d_{m\pm
1,\sigma}^{\ell}(\theta)\,J_{m-\sigma}\left(k\rho_{s}\sin\theta\right)e^{i\,k\cos\theta\,z_{s}}\,,$
(29)

where $k=n_{w}k_{0}$ is the wavenumber in the aqueous host medium. The
corresponding wavevector makes an angle $\theta$ with respect to the optical
$z$-axis.

## References

  * (1) G. H. Wagniére, On chirality and the universal asymmetry: reflections on image and mirror image, Wiley-VCH, Zurich, (2007).
  * (2) J. B. Pendry, A chiral route to negative refraction, Science 306, 1353 (2004).
  * (3) E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, Metamaterial with negative index due to chirality, Phys. Rev. B 79, 035407 (2009).
  * (4) J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, Gold helix photonic metamaterial as broadband circular polarizer, Science 325, 1513 (2009).
  * (5) Y. Tang and A. Cohen, Enhanced enantioselectivity in excitation of chiral molecules by superchiral light, Science 332, 333 (2011).
  * (6) M. J. Urban, C. Shen, X.-T. Kong, C. Zhu, A. O. Govorov, Q. Wang, M. Hentschel, N. Liu, Chiral plasmonic nanostructures enabled by bottom-up approaches, Ann. Rev. of Phy. Chem. 70, 275 (2019).
  * (7) M. Hentschel, M. Schaferling, X. Duan, H. Giessen, and N. Liu, Chiral plasmonics, Science Advances 3, e1602735 (2017).
  * (8) X.-T. Kong, L. V. Besteiro, Z. Wang, and A. O. Govorov, Plasmonic chirality and circular dichroism in bioassembled and nonbiological systems: theoretical background and recent progress Adv. Mater. 1801790 (2018).
  * (9) Z. Fan, A. O. Govorov, Plasmonic circular dichroism of chiral metal nanoparticle assemblies, Nano Lett. 10, 2580 (2010).
  * (10) F. A. Pinheiro, V. A. Fedotov, N. Papasimakis, and N. I. Zheludev, Spontaneous natural optical activity in disordered media, Phys. Rev. B 95, 220201(R) (2017).
  * (11) Y. Zhao and J. Dionne, Response to ”Comment on enantioselective optical trapping of chiral nanoparticles with plasmonic tweezers”, ACS Photonics 5, 2535 (2018).
  * (12) A. J. Mastroianni, S. A. Claridge, P. Alivisatos, Pyramidal and chiral groupings of gold nanocrystals sssembled using DNA scaffolds, J. Am. Chem. Soc. 131, 8455 (2009).
  * (13) R. Ali, F. A. Pinheiro, R.S. Dutra, F. S. S. Rosa, and P. A. Maia Neto, Enantioselective manipulation of single chiral nanoparticles using optical tweezers, Nanoscale, 12, 5031 (2020).
  * (14) S. B. Wang and C.T. Chan, Lateral optical force on chiral particles near a surface, Nat. Commun. 5, 3307 (2014).
  * (15) A. Hayat, J. B. Mueller, and F. Capasso, Lateral chirality-sorting optical forces, Proc. Natl. Acad. Sci. U.S.A. 112, 13190 (2015).
  * (16) A. Canaguier-Durand and C. Genet, Plasmonic lateral forces on chiral spheres, J. Opt. 18, 015007 (2016).
  * (17) H. Chen, C. Liang, S. Liu, and Z. Lin, Chirality sorting using two-wave-interference-induced lateral optical force, Phys. Rev. A 93, 053833 (2016).
  * (18) T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, All-optical chirality-sensitive sorting via reversible lateral forces in interference fields, ACS Nano 11, 4292 (2017).
  * (19) T. Cao and Y. Qiu, Lateral sorting of chiral nanoparticles using Fano-enhanced chiral force in visible region, Nanoscale 10, 566 (2018).
  * (20) M. H. Alizadeh and B. M. Reinhard, Enhanced optical chirality through locally excited surface plasmon polaritons, ACS Photonics 2, 1780 (2015).
  * (21) Y. Li, C. Bruder, and C. Sun, Generalized Stern-Gerlach effect for chiral molecules, Phys. Rev. Lett. 99, 130403 (2007).
  * (22) B. Spivak and A. Andreev, Photoinduced separation of chiral isomers in a classical buffer gas, Phys. Rev. Lett. 102, 063004 (2009).
  * (23) X. Li and M. Shapiro, Theory of the optical spatial separation of racemic mixtures of chiral molecules, J. Chem. Phys. 132, 194315 (2010).
  * (24) R. P. Cameron, S. M. Barnett, and A. M. Yao, Discriminatory optical force for chiral molecules, New J. Phys. 16, 013020 (2014).
  * (25) A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, Mechanical separation of chiral dipoles by chiral light, New J. Phys. 15, 123037 (2013).
  * (26) D. S. Bradshaw and D. L. Andrews, Chiral discrimination in optical trapping and manipulation, New J. Phys. 16, 103021 (2014).
  * (27) Y. Zhao, A. A. Saleh, and J. A. Dionne, Enantioselective optical trapping of chiral nanoparticles with plasmonic tweezers, ACS Photonics 3, 304 (2016).
  * (28) P. Acebal, L. Carretero, and S. Blaya, Design of an optical conveyor for selective separation of a mixture of enantiomers, Opt. Express 25, 32290 (2017).
  * (29) C.-S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. Dionne, Enhancing enantioselective absorption using dielectric nanospheres, ACS Photonics, 4, 197 (2017).
  * (30) M. L . Solomon, J. Hu, M Lawrence, A. García-Etxarri, J. A. Dionne, Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces, ACS Photonics 6 , 43 (2018).
  * (31) R. J. Hernandez, A. Mazzulla, A. Pane, K. Volke- Sepulveda, and G. Cipparrone, Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers, Lab Chip 13, 459 (2013).
  * (32) G. Tkachenko and E. Brasselet, Optofluidic sorting of material chirality by chiral light, Nat. Commun. 5, 3577 (2014).
  * (33) M. Donato, J. Hernandez, A. Mazzulla, C. Provenzano, R. Saija, R. Sayed, S. Vasi, A. Magazzu, P. Pagliusi, R. Bartolino, P. Gucciardi, O. Marago, and G. Cipparrone,  Polarization-dependent optomechanics mediated by chiral microresonators, Nat. Commun. 5, 3656 (2014).
  * (34) G. Tkachenko and E. Brasselet, Helicity-dependent three-dimensional optical trapping of chiral microparticles, Nat. Commun. 5, 4491 (2014).
  * (35) Y Zhao et al., Nanoscopic control and quantification of enantioselective optical forces, Nat. nanotechnology 12, 1055 (2017).
  * (36) G. Schnoering, L. V. Poulikakos, Y. Rosales-Cabara, A. Canaguier-Durand, D. J. Norris, and C. Genet, Three-dimensional enantiomeric recognition of optically trapped single chiral nanoparticles, Phys. Rev. Lett. 121, 023902 (2018).
  * (37) N. Kravets, A. Aleksanyan, and E. Brasselet, Chiral optical Stern-Gerlach Newtonian experiment, Phys. Rev. Lett. 122, 024301 (2019).
  * (38) N. Kravets, A. Aleksanyan, H. Chraibi, J. Leng, and E. Brasselet, Optical enantioseparation of racemic emulsions of chiral microparticles, Phys. Rev. App. 11, 044025 (2019).
  * (39) Z. Fan, A. O. Govorov, Chiral nanocrystals: plasmonic spectra and circular dichroism, Nano Lett. 12, 3283 (2012).
  * (40) P. A. Maia Neto and H. M. Nussenzveig, Theory of optical tweezers, Europhys. Lett. 50, 702 (2000).
  * (41) A. Mazolli, P. A. Maia Neto, H. M. Nussenzveig, Theory of trapping forces in optical tweezers, Proc. R. Soc. Lond. A, 459, 3041 (2003).
  * (42) V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic waves in chiral and bi-isotropic media, (Artech House, Boston, 1994).
  * (43) A. Lakhtakia and V. V. Varadan, Time-harmonic electromagnetic fields in chiral media, (Springer-Verlag, Berlin, 1989).
  * (44) H. Chen, N. Wang, Wanli L,, Shiyang Liu, and Zhifang Lin, Tailoring azimuthal optical force on lossy chiral particles in Bessel beams, Phys. Rev. A, 90, 043850 (2014).
  * (45) If $\epsilon\mu=\kappa^{2}$ we have the chiral analogue of bulk plasmons, which are longitudinal solutions for the fields.
  * (46) C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1998).
  * (47) M. Born and E. Wolf, Principles of Optics,(Pergamon, Oxford, 1980).
  * (48) C.R. Bohren, Light scattering by an optically active sphere, Chem. Phys. Lett. 29, 458 (1974).
  * (49) Q. Shang, Z. Wu, Tan Qu, Z. Li, Lu Bai, and L. Gong,  Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam, Opt. Exp. 21, 8677 (2013).
  * (50) M. Abramowitz and I. Stegun, Handbook of mathematical functions, (Dover, New York, 1972).
  * (51) B. Richards and E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system, Proc. R. Soc. London A 253, 358 (1959).
  * (52) R. S. Dutra, N. B. Viana, P. A. Maia Neto and H. M. Nussenzveig, Absolute calibration of forces in optical tweezers, Phys. Rev. A 90, 013825 (2014).
  * (53) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, 1957).
  * (54) K. Diniz, R. S. Dutra, L. B. Pires, N. B. Viana, H. M. Nussenzveig, and P. A. Maia Neto,  Negative optical torque on a microsphere in optical tweezers, Opt. Express 27, 5905 (2019).
  * (55) D. S. Bradshaw and D. L Andrews, Manipulating particles with light: radiation and gradient forces, Eur. J. Phys. 38, 034008 (2017).