# Ultraviolet Mie resonances in computationally discovered boron phosphide
nanoparticles

Mark Kamper Svendsen  [  Hiroshi Sugimoto  [  Artyom Assadillayev  [  Daisuke
Shima  [  Minoru Fujii  [  Kristian S. Thygesen  [<EMAIL_ADDRESS>Søren Raza  [<EMAIL_ADDRESS>
###### Abstract

Controlling ultraviolet light at the nanoscale using optical Mie resonances
holds great promise for a diverse set of applications, such as lithography,
sterilization, and biospectroscopy. However, Mie resonances hosted by
dielectric nanoantennas are difficult to realize at ultraviolet wavelengths
due to the lack of both suitable materials and fabrication methods. Here, we
systematically search for improved materials by computing the frequency
dependent optical permittivity of 338 binary semiconductors and insulators
from first principles, and evaluate their potential performance as high
refractive index materials using Mie theory. Our analysis reveals several
interesting candidate materials among which boron phosphide (BP) appears
particularly promising. We then prepare BP nanoparticles and demonstrate that
they support Mie resonances at visible and ultraviolet wavelengths using both
far-field optical measurements and near-field electron energy-loss
spectroscopy. We also present a laser reshaping method to realize spherical
Mie-resonant BP nanoparticles. With a refractive index above 3 and low
absorption losses, BP nanostructures advance Mie optics to the ultraviolet.

DTU] Department of Physics, Technical University of Denmark, Fysikvej, DK-2800
Kongens Lyngby, Denmark Kobe] Department of Electrical and Electronic
Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan DTU]
Department of Physics, Technical University of Denmark, Fysikvej, DK-2800
Kongens Lyngby, Denmark Kobe] Department of Electrical and Electronic
Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan Kobe]
Department of Electrical and Electronic Engineering, Kobe University,
Rokkodai, Nada, Kobe 657-8501, Japan DTU] Department of Physics, Technical
University of Denmark, Fysikvej, DK-2800 Kongens Lyngby, Denmark DTU]
Department of Physics, Technical University of Denmark, Fysikvej, DK-2800
Kongens Lyngby, Denmark

## Introduction

Achieving control over ultraviolet light with nanoscale materials is essential
for improving surface-enhanced spectroscopies of biological molecules and
enabling new ultraviolet optical components. Geometric Mie resonances
supported by resonant nanoantennas made from materials that combine a high
refractive index with low absorption losses offer efficient and tunable
manipulation of the near- and far-field of optical waves1, 2. Mie resonances
have been realized at visible and infrared wavelengths thanks to the mature
lithographic processing of suitable materials3, such as silicon4, gallium
phosphide5, and titanium dioxide6. It would be desirable to extend the
operation of these materials to the ultraviolet, but their small direct band
gap energies ($\lesssim 3$ eV) lead to significant absorption losses in the
ultraviolet. Recently, metasurfaces composed of an array of nanostructured
materials with a wide band gap and moderate refractive index ($n\approx
2.1-2.3$) have demonstrated wave front manipulation in the ultraviolet using
waveguide modes7, 8, 9. For ultraviolet Mie optics, diamond has been
theoretically suggested as a potential material 10, 11, but diamond comes with
significant nanofabrication challenges12. Consequently, extending the rich
optical properties of Mie resonances observed in the visible to the
ultraviolet requires identification of new high-index materials as well as
development of suitable fabrication methods to realize Mie-resonant
nanoantennas.

Concurrent advances in first-principles methodology and computing power have
recently made it possible to design and discover new materials via high-
throughput computations13, 14, 15, 16, 17. The approach has been successfully
applied in several domains, including photovoltaics, transparent conductors,
and photocatalysis18, 19, 20. However, to the best of our knowledge,
computational discovery of new high-index materials remains largely
unexplored. Relevant previous work in this direction has been limited to the
static response regime21, 22 reflecting the fact that the major materials
databases so far has focused on ground state properties.

Here we use high-throughput linear response density functional theory (DFT) to
screen an initial set of 2743 elementary and binary materials with the aim to
identify isotropic high-index, low loss, and broad band optical materials. For
the most promising materials, the computed frequency-dependent complex
refractive indices are used as input for Mie scattering calculations to
evaluate their optical performance. In addition to the already known high-
index materials we identify several new compounds. In particular, boron
phosphide (BP) offers a refractive index above 3 with very low absorption
losses in a spectral range spanning from the infrared to the ultraviolet. We
then prepare BP nanoparticles and show, by means of dark-field optical
measurements and electron energy-loss spectroscopy, that they support size-
dependent Mie resonances in the visible and ultraviolet. Finally, we
demonstrate a laser reshaping method to realize spherical BP nanoparticles,
which host multiple Mie resonances in quantitative agreement with full-field
optical simulations.

## Results

### High-throughput screening

Our high-throughput screening procedure is illustrated in Fig. 1a. We build
the workflow using the Python-based Atomic Simulation Recipes (ASR)23
framework and the MyQueue scheduling software24 (see detailed workflow in
Supplementary Note 1 and Supplementary Fig. 1). Starting from 2743
thermodynamically stable elementary and binary materials from the Open Quantum
Materials Database (OQMD)15 we extract the 1693 materials with up to 10 atoms
in the unit cell and relax the atomic structure using DFT with the Perdew-
Burke-Ernzerhof (PBE) exchange correlation functional25 and the D3 correction
to account for the van der Waals forces26. We perform DFT ground state
calculations for all of the materials to determine their electronic band gaps.
After discarding the metals, we are left with 338 semiconductors for which we
calculate the optical dielectric function, $\epsilon(\omega)$, within the
random phase approximation (RPA) and extract the refractive index and
extinction coefficient as
$n(\omega)=\mathrm{Re}\left\\{\sqrt{\varepsilon(\omega)}\right\\}$ and
$k(\omega)=\mathrm{Im}\left\\{\sqrt{\varepsilon(\omega)}\right\\}$,
respectively. All calculations are performed with the GPAW code27, 28.

Figure 1: High-throughput materials screening. a, Schematic of the screening
steps and the number of materials that survive each of them. b, The static
refractive index of the isotropic materials plotted as a function of their
direct band gap energies along with the Moss formula. The colored dots
highlight interesting materials and the inset shows the energy-dependent
$Q$-factor and radiation efficiency, $\eta$, of the MD resonances in those
materials. The background colors represent the infrared (red), visible (green)
and ultraviolet (blue) spectral regions.

Next, we classify the materials according to the anisotropy of their
refractive index tensor using a cut-off of 0.05 for the fractional anisotropy
(see Supplementary Fig. 2). This leaves us with 207 isotropic material
candidates, for which we show the static refractive index as a function of the
direct band gap in Fig. 1b. The data points qualitatively follow the Moss
formula29 (dashed line in Fig. 1b); however, there are significant deviations
from the general trend, which we ascribe to variations in oscillator strength
and density of the transitions across the direct gap. It is well known that
(semi)local functionals like the PBE employed in the present work
systematically underestimates band gaps30. This effect is, however, to some
extent compensated by the fact that the RPA neglects the attractive electron-
hole interactions and consequently underestimates the spectral weight near the
band edge. As a result, refractive indices obtained with RPA@PBE are typically
in good agreement with experiments in the static limit28 while deviations
occur at higher frequencies near the band edge region (see Supplementary Note
2 and Supplementary Fig. 3-5). We shall return to this point later, but for
now mention that for the materials that are identified as interesting based on
this screening we will employ more accurate and computational expensive many-
body perturbation methods.

We now turn to a more in-depth evaluation of the performance of the discovered
materials. Specifically, we use Mie theory to calculate the scattering
properties of a spherical nanoparticle made from the subset of isotropic
materials. We focus on the lowest-order magnetic dipole (MD) resonance of the
spheres and calculate energy-dependent quality factors, $Q$, and radiation
efficiencies, $\eta$ (see Supplementary Note 1)3. This is achieved by
continuously adjusting the size of the sphere to tune the energy of the MD
resonance across the infrared, visible and ultraviolet regions. A high
$Q$-factor is beneficial for boosting the local field enhancement of the
nanoparticle, while a radiation efficiency close to unity points to low
absorption losses. This analysis identifies all of the commonly used
materials, such as Si, TiO2, and GaP. However, we also find a number of other
highly promising materials, some of which are highlighted in the inset of Fig.
1b. In particular, we identify BP, which has a refractive index exceeding that
of TiO2 and a radiation efficiency higher than both silicon and TiO2 across
the entire visible part of the spectrum. For these reasons, we believe BP
stands out as an overlooked material with highly desirable optical properties
and we will focus on BP in the rest of the paper.

### Optical response of BP

Figure 2: Optical response and Mie resonances of BP. a,b Refractive index $n$
and extinction coefficient $k$ of BP calculated using the f-sum corrected BSE
method compared with that of some of the commonly used dielectrics in the
visible and ultraviolet spectral regions. c, Extinction efficiency map of BP
spheres of varying radii calculated using Mie theory, which demonstrates that
Mie resonances can be sustained in the visible and ultraviolet.

BP crystals were successfully synthesized as early as 195731, yet experimental
measurements of its refractive index are limited to a couple of data points in
the visible32, 33. Refractive index measurements of BP thin films have also
been conducted but with varying results34, 35.

The RPA@PBE permittivities used for the initial screening are qualitatively
accurate but suffer from underestimated band gaps and missing excitonic
effects. To determine the refractive index of BP with quantitative accuracy,
we solve the Bethe-Salpeter equation (BSE) to obtain the permittivity using
single-particle transition energies obtained from a G0W0 band structure
calculation. The band structure calculation reveals an indirect band gap of
2.1 eV and a direct band gap of 4.41 eV, which matches experimental
measurements of the band gap energies for BP.36, 37 The square root dependence
of the refractive index on the permittivity makes it crucial to converge both
the real and imaginary parts of the latter. Unfortunately, the real part
converges slowly with the number of bands making it impractical to obtain well
converged results directly from the BSE. The problem can be alleviated by
extending the imaginary part of the permittivity by an exponentially decaying
tail whose weight is fixed by the f-sum rule (see Methods), and subsequently
obtain the real part via the Kramers–Kronig relation. We benchmark this
approach against experimental data for the refractive index of crystalline
silicon and find excellent agreement (see Supplementary Note 3 and
Supplementary Fig. 6).

With the f-sum rule fulfilling BSE-G0W0 method at hand, we are in a position
to make a quantitative comparison of the refractive index of BP with some of
the commonly used materials38, 39 in the visible as well as diamond40, which
has been theoretically suggested for operation in the ultraviolet (Fig. 2a,b).
We observe that the absorption edge of BP lies significantly higher than
silicon, GaP, and TiO2, while it retains a refractive index comparable to that
of GaP. This suggests that BP provides low-loss operation across the entire
visible spectrum. While this is also the case for TiO2, its refractive index
is significantly lower than that of BP. The higher refractive index of BP
means that nanostructures can be made more compact41 and packed more densely
for enhanced metasurface performance42. Importantly, we note that BP offers a
high refractive index with a low extinction coefficient not only in the
visible but also in the ultraviolet – a spectral region which is unreachable
with the commonly-used materials. Diamond is transparent in the ultraviolet as
well, but has a significantly lower refractive index. We illustrate the
broadband performance of BP by performing extinction efficiency calculations
using Mie theory for a BP sphere with varying radii, confirming that Mie
resonances can be sustained across the visible and ultraviolet (Fig. 2c).

Figure 3: Dark-field scattering of BP nanoparticles. a, Schematic of the dark-
field scattering spectroscopy. The nanoparticles are excited by unpolarized,
white light incident at an oblique angle and the transmitted scattered light
is collected. b, Scattering efficiency spectra recorded from individual BP
nanoparticles of different sizes showing clear scattering peaks due to
resonant interaction. The effective radius $R$ of the nanoparticle is
determined from the area of the particles, assuming a spherical shape. c,
Simulated scattering efficiency map of BP nanospheres with varying particle
radii.

### Far- and near-field characterization of BP nanoparticles

We now turn to an experimental demonstration of the potential of BP as a Mie-
resonant nanostructure. BP has been synthesized in various forms such as
crystals43, 44, films35, 45, 46, and nanoparticles47, 48, 49. However, BP
nanoparticles in the size range suitable for sustaining Mie resonances have
not been reported. We prepare BP nanoparticles in the size range of a few
hundred nanometers by grounding BP powder (Kojundo Chemicals) in a mortar and
then dispersing it in methanol. The BP solution is subsequently dropcasted on
a glass substrate.

We measure the far-field scattering efficiency of individual nanoparticles by
illuminating the nanoparticles through a high-numerical-aperture dark-field
objective and collecting the transmitted scattered light (see Fig. 3a). A
similar measurement setup has been used to detect Mie resonances in silicon
nanoparticles.50 The scattering spectra recorded from a series of BP
nanoparticles clearly show resonant peaks, which red shift with increasing
particle size (Fig. 3b). Despite the irregular particle shapes, the scattering
resonances are quite prominent and follow the trend observed in other Mie-
resonant nanostructures, namely, that the smallest particle size support only
the lowest-order Mie resonance while larger particle sizes also support
higher-order Mie resonances51. We additionally confirm the crystallinity of
the nanoparticles using micro-Raman spectroscopy from individual nanoparticles
(see Supplementary Fig. 7). The particle radii are extracted from the scanning
electron microscopy images under the assumption of a spherical shape and used
as input for full-field simulations of the scattering efficiency of a BP
nanosphere. The simulations account for the measurement setup as well as the
glass substrate (see Methods). We find that the simulations accurately
reproduce both the shift in resonance wavelengths with particle size as well
as the number of resonant peaks (Fig. 3c). However, the particle sizes need to
be adjusted to match the resonance wavelengths observed in the experiments.
This suggests that the particle shape is better characterized as flakes with a
thickness significantly smaller than the in-plane size. Nonetheless, the
distinctive, multiple scattering peaks provide strong evidence for the
interpretation that these are related to geometric Mie resonances.

Figure 4: Electron energy-loss spectroscopy of BP nanoparticles. a,b
Experimental STEM image and theoretical set-up depicting the electron beam
positions $b$ used to acquire the measured and simulated EELS spectra of c,d,
respectively. The triangular-shaped BP nanoparticle is characterized by an
effective radius of $R=90$ nm. Several EELS peaks due to Mie resonances are
observed. e-g Simulated and h-j experimental EELS intensity maps at the Mie
resonance energies. The Mie modes are identified as the in-plane electric
dipole (ED∥), in-plane electric quadrupole (EQ∥), and out-of-plane electric
dipole (ED⟂) using a multipole decomposition.

To gain more insight into the nature of these resonances and to access the
ultraviolet spectral region, we also perform near-field characterization on
similar BP nanoparticles using electron energy-loss spectroscopy (EELS). EELS
is performed in a transmission electron microscope and has been employed to
access near-field properties of both metallic52, 53 and dielectric
nanostructures54, 55, 56 as well as optical devices57. The combined high
spatial and spectral resolution of EELS provides unique nanoscale information
on optical modes over a broad spectral range. For EELS measurements, the BP
nanoparticles are deposited on a thin silicon oxide membrane. The EELS signal
recorded from a triangular-shaped BP nanoparticle (Fig. 4a) at different beam
positions is presented in Fig. 4c. The position of the beam is directly
related to the excitation efficiency of the optical modes 58, and thus, by
judicious positioning of the beam we can selectively excite different Mie
modes. 54 When the beam is positioned in the center of nanoparticle, we
observe a distinct resonance in the ultraviolet at 3.99 eV. As the beam is
moved closer to the surface of the nanoparticle, two additional resonances are
observed at the energies 3.57 eV and 2.77 eV. To identify the nature of these
resonances, we simulate the EELS signal of a BP nanodisk with the same
effective radius as the measured BP nanoparticle (Fig. 4b). The thickness of
the nanodisk is varied to achieve correspondence to the measured EELS
resonance energies. The simulated EELS spectra for a nanodisk thickness of
$h=40$ nm at the same beam positions as in the experiments are shown in Fig.
4d. Here, we observe that the three lowest-energy EELS peaks have the same
dependence on the impact parameter of the electron beam as seen
experimentally. The simulated resonance energies of all three EELS peaks are
also in quantitative agreement with the experiments, albeit the lowest-energy
EELS peak is slightly shifted to higher energies in the simulations. The
measured EELS peaks are broadened by the finite energy resolution of our EELS
setup (see Methods). By performing a multipole decomposition of the induced
field produced by the electron beam59, we identify the two lowest energy EELS
peaks to be due to the Mie modes of the in-plane electric dipole ED∥ and in-
plane electric quadrupole EQ∥. The highest energy EELS peak has contributions
from both the in-plane magnetic dipole and and out-of-plane electric dipole
ED⟂, where the latter dominates in the center of the particle (see
Supplementary Fig. 8 for full decomposition). Simulated EELS intensity maps
show that the electron beam couples efficiently to the in-plane modes, ED∥ and
EQ∥ for beam positions near the surface of the particle, while the out-of-
plane ED⟂ is excited also for beam positions in the center of the particle
(Fig. 4e-g). The experimental EELS intensity maps of these three Mie modes are
in good agreement with the simulations (Fig. 4h-j) as well as previous EELS
measurements performed on Mie-resonant silicon nanoparticles54, 55. The near-
field EELS measurements along with far-field dark-field optical measurements
demonstrate that, despite their irregular shape, BP nanoparticles host a
variety of multipolar size-dependent Mie resonances across the visible and
ultraviolet, which are key attributes of low-loss high-index nanostructures.

### Laser reshaping

To alleviate the irregular shape of the as-prepared BP nanoparticles, we
generate spherical BP nanoparticles by irradiating the unprocessed BP
nanoparticles with a pulsed laser (see Fig. 5a and Methods). An example of a
laser-processed BP nanoparticle is presented in Fig. 5b, confirming that the
laser processing can be used for realizing spherical-shaped BP nanoparticles
without affecting the nanoparticle composition. In addition, we perform micro-
Raman spectroscopy on the same particle and find that the crystallinity from
the unprocessed BP nanoparticles is retained after laser reshaping (see
Supplementary Fig. 7). The dark-field scattering spectrum from the BP
nanoparticle in Fig. 5b is recorded and we observe multiple scattering peaks
(Fig. 5c). The relatively large particle radius ($R=184$ nm) places the
lowest-order Mie resonances at wavelengths longer than our measurement range,
while the scattering peaks observed can be attributed to higher-order Mie
resonances. The full-field simulation of a BP nanosphere with a slightly
smaller radius $R=165$ nm, where we account for the substrate and the
measurement setup, shows very good agreement with the measurement. We
attribute the deviation in particle radius to shape imperfections and a slight
variation between the calculated and experimental refractive index of BP.
Multipole decomposition reveals that the scattering peaks are due to the
excitation of the magnetic quadrupole MQ, the electric quadrupole EQ, and
radial higher-order magnetic dipole MD2, thereby confirming the Mie-resonant
nature of BP nanoparticles. We also performed EELS measurements on a smaller
laser-reshaped BP nanoparticle, where we observe Mie resonances in the
ultraviolet (see Supplementary Fig. 9).

Figure 5: Laser reshaping of BP nanoparticles. a, Reshaping of the BP
nanoparticles is achieved by pulsed laser irradiation of unprocessed BP
nanoparticles in acetone. After the irradiation, the BP nanoparticles obtain a
spherical shape. b, Scanning electron microscopy image and energy-dispersive
x-ray analysis of reshaped BP nanoparticle, showing a spherical shape without
change in composition. c, Scattering spectrum recorded from the BP
nanoparticle in b and simulated spectrum for a BP nanosphere with radius
$R=165$ nm with multipole decomposition revealing the excitation of the
magnetic quadrupole (MQ), electric quadrupole (EQ), and radial higher-order
magnetic dipole (MD2).

## Discussion

Using a DFT-based high-throughput screening method combined with optical Mie
theory of 338 dielectrics, we identify BP as a promising high-refractive-index
material for ultraviolet nanooptics. We develop a new, quantitatively accurate
many-body perturbation theory based methodology for calculating refractive
indices and use it to reveal that BP has a high refractive index ($n>3$) and
low extinction coefficient ($k<0.1$) up to ultraviolet photon energies of 4
eV. We present an approach to fabricate BP nanoparticles as well as a laser
reshaping method to generate spherical BP nanoparticles. Through dark-field
optical spectroscopy and EELS measurements, we confirm the presence of Mie
resonances in BP nanoparticles across the visible and ultraviolet. Our work
advances nanoscale Mie optics to the ultraviolet and may find applications in
metasurface-enhanced spectroscopy of biological molecules and, more generally,
in realizing metasurface optical components operating in the ultraviolet.

## Methods

### Computational workflow

#### Structural relaxation:

All ground- and excited state calculations were performed with the GPAW
electronic structure code27. We relaxed the atomic structure and the unit cell
of the materials until the maximum force (stress) is below $10^{-4}$ eV/Å
(0.002eV/Å3). We use the PBE functional for exchange and correlation effects,
a $\Gamma$-point centered k-point grid with a density of $6.0$Å-3, a 800 eV
plane wave cutoff and a Fermi-Dirac smearing of 50 meV. Van der Waals
interactions were taken into account by the D3 correction scheme26.

#### RPA calculations:

We calculate the optical permittivity, $\varepsilon(\omega)$, within the
Random Phase Approximation(RPA) using the dielectric function module in GPAW.
From $\epsilon(\omega)$ we calculate the refractive index and extinction
coefficient as
$n(\omega)=\mathrm{Re}\left\\{\sqrt{\varepsilon(\omega)}\right\\}$ and
$k(\omega)=\mathrm{Im}\left\\{\sqrt{\varepsilon(\omega)}\right\\}$
respectively. To ensure convergence across all materials we employ a k-point
grid with a high density of 20.0 Å-3 and include conduction bands up to 5
times the number of valence bands. The calculations were performed on a
nonlinear frequency grid with an initial frequency spacing of 0.5 meV, a
broadening of 50 meV and a local field cutoff of 50 eV.

### Many-body perturbation theory calculations

#### G0W0 calculations:

The G0W0 calculations were performed on top of the ground state calculations.
To ensure converged quasi-particle gaps we perform extrapolation of both the
plane wave cut-off and the k-point resolution to infinity.

#### BSE calculations:

The BSE calculations were performed within the Tamm-Dancoff approximation on a
k-point grid with a density of 20 Å-3. The calculation included all valence
and conduction bands within 2.3 eV of the valence band maximum and conduction
minimum, respectively. The calculation of the screened interaction included
all occupied bands and unoccupied bands up to 5 times the number of occupied
bands, and we accounted for local field effects up to a plane wave cut-off of
50 eV. The calculation was performed on a linear 10001 point frequency grid
spanning 0 to 8 eV.

#### BSE f-sum rule correction

The optical polarizability has to obey the f-sum rule,

$\displaystyle\int_{0}^{\infty}d\omega\omega\mathrm{Im}\chi(\omega)=\frac{\pi}{2}\frac{n_{e}e^{2}}{m}\,,$
(1)

where $n_{e}$ is the electron density, $e$ is the elementary charge and $m$ is
the electron mass. Since it is not obvious what to use for $n_{e}$ in our PAW
calculations, we use a different strategy to fix the left hand side, namely we
obtain it from an RPA calculation (which can be readily converged)

$\displaystyle\int_{0}^{\infty}d\omega\omega\mathrm{Im}\chi^{\mathrm{(RPA)}}(\omega)=\int_{0}^{\infty}d\omega\omega\mathrm{Im}\chi^{\mathrm{(BSE)}}(\omega)\,.$
(2)

Eq. 2 can be enforced if we artificially extend the imaginary part of the BSE
polarizability. Denoting the largest transition energy included in the BSE
calculation as $w_{c}$, we perform the following extension,

$\displaystyle\mathrm{Im}\chi^{\mathrm{(BSE)}}(\omega)=\begin{cases}\mathrm{Im}\chi^{\mathrm{(BSE)}}(\omega),&\text{if
}\omega\leq\omega_{c},\\\ C_{0}e^{-\gamma(\omega-\omega_{c})},&\text{if
}\omega>\omega_{c}.\end{cases}$ (3)

The constant $C_{0}$ is used to ensure continuity and the constant $\gamma$ is
fixed to give the correct spectral weight as fixed by Eq. 2. For benchmarks
see Supplementary Information.

### Optical simulations

The scattering efficiency and EELS simulations are both performed in COMSOL
Multiphysics (ver. 5.6), which solves Maxwell’s equations using finite-element
modelling. For the scattering efficiency simulations in Fig. 3c, a BP sphere
is placed on a semi-infinite glass substrate ($n_{\textrm{sub}}=1.45$) and
excited by a plane wave incident from the substrate side at an oblique angle
of $\theta_{\textrm{inc}}=42^{\circ}$. We use the f-sum corrected refractive
index for BP shown in Fig. 2. We then perform a near-to-far field
transformation to extract the scattered far-field60. The Poynting flux of the
scattered far-field in the air region is integrated over a solid angle
spanning an azimuthal angle of $2\pi$ and a maximum polar angle of
$\theta_{\textrm{col}}=\arcsin(\textrm{NA})=53^{\circ}$ to retrieve the total
scattered power collected by the collection objective ($\textrm{NA}=0.8$). The
total scattered power is normalized to the incident power and the geometrical
cross sectional area of the particle to determine the scattering efficiency.
To account for the unpolarized incident light in the experiments, we perform
this simulation procedure for both transverse-electric and transverse-magnetic
polarization of the incident wave. Finally, the scattering efficiency from
both polarization states is averaged. The scattering efficiency simulation in
Fig. 5c follows the same steps with the only change being that the plane wave
is incident from the air side at an oblique angle of
$\theta_{\textrm{inc}}=75^{\circ}$.

For the EELS calculations, we simulate the electron beam as an edge current
with an amplitude of 1 µA. The induced electromagnetic field is obtained by
calculating the fields with and without the BP nanodisk in the simulation
domain, and subsequently subtracting them. The energy loss is then calculated
as the work rate done on the electron beam by the induced electromagnetic
field 54.

### Dark-field scattering measurements

A custom-built inverted optical microscope was used for dark-field scattering
spectroscopy of single nanoparticles (Fig. 3a). For the measurements presented
in Fig. 3, the sample is illuminated from the top by a halogen lamp through a
dark field condenser and the scattered light was collected by an objective
(50×, $\textrm{NA}=0.8$). For the spectrum in Fig. 5c, the sample was
illuminated from the bottom through a dark-field objective (50×,
$\textrm{NA}=0.8$) and the scattered light was collected by the same
objective. To measure the spectra, scattered light was transferred to the
entrance slit of a monochromator (SpectraPro-300i, Princeton Instruments) and
detected by a liquid-N2 cooled CCD (Princeton Instruments). For Raman
scattering measurements, the nanoparticles were excited by a 488 nm laser
(Coherent Sapphire 488–50).

### EELS measurements and analysis

The EELS measurements are performed in a monochromated and aberration-
corrected FEI Titan operated in STEM mode at an acceleration voltage of 300
kV, providing a probe size of ${\sim}0.5$ nm and an energy resolution of 0.08
eV (as measured by the full-width-at-half-maximum of the zero-loss peak). We
perform Richardson–Lucy deconvolution to remove the zero-loss peak. An EELS
spectrum recorded in vacuum is used as an input for the point-spread function.
Due to a small asymmetry in the zero-loss peak, the deconvolution algorithm
produced an artificial EELS peak in the energy range below 0.5 eV. However,
the artificial peak did not overlap with any of the observed resonances and
could be safely removed using a first-order logarithmic polynomial.

The depicted EELS spectra are obtained by integrating the deconvoluted EELS
data around the experimental impact parameter $b_{\textrm{exp}}$. This
parameter is directly related to the effective radius of the particle $R$
which is extracted as a radius of the circle with the effective area of the
particle found from the STEM image. The effective area of the particle is the
area which is enclosed by the boundaries obtained by Otsu’s thresholding of
the STEM image. The integration parameter itself is calculated as a radius of
the circle with the adjusted effective area of expanded/reduced initial
boundaries. For the integration region centered at the nanoparticle, the
experimental impact parameter changes from 0 to 0.3$R$ and denotes the
nanoparticle center. For the annulus-shaped regions, the experimental impact
parameter denotes the mean of the inner and outer radii with a typical radius
spread of 0.3 (for example, the green region in Figure 4(a) encloses the
regions from 0.35 to 0.65$R$). The depicted EELS spectra are smoothed with a
Gaussian function ($\sigma=0.03$ eV).

The EELS maps are obtained by summing the deconvoluted EELS data in a spectral
window of $0.02$ eV width centered at the resonance energies. The signal-to-
noise ratio is improved by spatially binning the map, reducing the total
number of pixels by a factor of 4. A Gaussian filter with $\sigma=0.8$ pixels
is applied to smooth the image.

### Laser reshaping

Unprocessed BP nanoparticles in acetone are irradiated with the third harmonic
of a Nd:YAG laser (355 nm wavelength, 5 nm pulse width, 20 Hz repetition rate)
with a fluence of 50 mJ/cm2 per pulse for 10 min. S. R. and A. A. acknowledge
support from the Independent Research Funding Denmark (7026-00117B). K. S. T.
acknowledges support from the Center for Nanostructured Graphene (CNG) under
the Danish National Research Foundation (project DNRF103) and from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant No. 773122, LIMA). K. S. T. is a Villum
Investigator supported by VILLUM FONDEN (grant no. 37789). H. S. acknowledges
support by JSPS KAKENHI Grant Numbers 21K14496.

Author contributions M. K. S. constructed the computational workflow,
performed the screening, developed the f-sum approach, and prepared the
figures. H. S. performed the structural and optical characterizations of
nanoparticles. A. A. performed the EELS measurements, EELS analysis, EELS
simulations, and prepared a figure. D. S. fabricated the nanoparticles and
conducted laser shaping processes. M. F. contributed to analyses and
interpretation of the data and supervised the fabrication and
characterizations of nanoparticles. S. R. and K. S. T. conceived the idea and
supervised the work. S. R. performed the optical simulations. All authors
discussed the results and contributed to the preparation of the manuscript.

Conflict of interests The authors declare no competing financial interests.

Data and materials availability All data needed to evaluate the conclusions in
the paper are present in the paper and/or the Supplementary Information.
Additional data related to this paper may be requested from the corresponding
author.

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