# Optical Materials for Maximal Nanophotonic Response

Hyungki Shim  Department of Applied Physics, Yale University, New Haven,
Connecticut 06511, USA  Energy Sciences Institute, Yale University, New Haven,
Connecticut 06511, USA  Department of Physics, Yale University, New Haven,
Connecticut 06511, USA  Zeyu Kuang  Department of Applied Physics, Yale
University, New Haven, Connecticut 06511, USA  Energy Sciences Institute, Yale
University, New Haven, Connecticut 06511, USA  Owen D. Miller  Department of
Applied Physics, Yale University, New Haven, Connecticut 06511, USA  Energy
Sciences Institute, Yale University, New Haven, Connecticut 06511, USA

###### Abstract

This article reviews the material properties that enable maximum optical
response. We highlight theoretical results that enable shape-independent
quantification of material “figures of merit,” ranging from classical sum
rules to more recent single-frequency scattering bounds. A key delineation at
optical frequencies is between polaritonic materials that support highly
subwavelength resonances and dielectric materials that can have vanishingly
small loss rates. We discuss the key metrics that enable comparisons both
within these material classes and between them. We discuss analogous metrics
for 2D materials, and point to applications for which rigorous comparison can
be made between bulk- and 2D-material approaches. The various results
highlight the synergy between materials discovery and theoretical nanophotonic
bounds, and point to opportunities in achieving new extremes in light–matter
interactions.

##  I Introduction

As material discovery proceeds at a rapid pace Goldberger  _et al._ (2003);
Yaghi  _et al._ (2003); van Hest  _et al._ (2005); Macak  _et al._ (2005);
Novoselov  _et al._ (2005); Geim and Novoselov (2007); Ayari  _et al._ (2007);
Paciĺ  _et al._ (2008); Hsieh  _et al._ (2008); Dean  _et al._ (2010); Mak
_et al._ (2010); Moore (2010); Osada and Sasaki (2012); Wang  _et al._
(2012a); Geim and Grigorieva (2013); Liu  _et al._ (2014a); Borisenko  _et
al._ (2014); Lv  _et al._ (2015); Saidaminov  _et al._ (2015); Bhimanapati
_et al._ (2015); Tan  _et al._ (2017), and atomistic control promises the
possibility of “designer” materials Frenkel (2002); Barredo  _et al._ (2016);
Kim  _et al._ (2016); Barredo  _et al._ (2018), there is a fundamental
question to be answered: what material properties should the optical-materials
community aim to synthesize? In the field of nanophotonics, for many
applications the goal is to maximize the interaction of light with matter,
manifested by absorption and/or scattering Gobin  _et al._ (2007); Huang  _et
al._ (2009); Anker  _et al._ (2008); Chang  _et al._ (2010); Pu  _et al._
(2010); Hsu  _et al._ (2014), quality factor Noda  _et al._ (2000); Michler
_et al._ (2000); Lončar  _et al._ (2003); Reithmaier  _et al._ (2004); Tanabe
_et al._ (2005); Hennessy  _et al._ (2007); Srinivasan and Painter (2007),
spontaneous-emission rate Nie and Emory (1997); Bardhan  _et al._ (2009);
Callahan  _et al._ (2012); Eggleston and Wu (2015), and related response
functions Laroche  _et al._ (2006); Martín-Cano  _et al._ (2010); Gonzalez-
Tudela  _et al._ (2011); Basu  _et al._ (2009), for frequency bands ranging
from the near ultraviolet to the far infrared. In this Review, we survey the
key metrics that have been identified for maximum optical response. Across the
broad landscape of polaritonic, dielectric, and 2D materials, we use
experimental optical-constant data in tandem with these metrics to identify
especially promising materials and material characteristics at optical
frequencies.

At any frequency, one can divide the landscape of nonmagnetic materials into
two categories: those whose permittivities have negative real parts, thereby
supporting quasistatic plasmonic and polaritonic resonances (with caveats
discussed below), and those with positive real parts. Following standard
terminology, we call the former “polaritonic” materials and the latter
“dielectric” materials, though by this definition the category a physical
material belongs to often changes with the frequency of interest. The reason
for this delineation is the significant differences in the resonator
properties of the two materials. For polaritonic materials (Sec. II), it is
possible and typically desirable to support quasistatic resonances with
extremely subwavelength confinement of electromagnetic waves Maier (2007);
Ozbay (2006); Caldwell  _et al._ (2015); Khurgin (2015), with a length scale
decoupled from the free-space wavelength and a resonant frequency determined
by the material permittivity and depolarization factor associated with the
shape. The dominant loss mechanism is dissipation (absorption), and for
materials with electric susceptibilities $\chi(\omega)$, there are two key
metrics pertaining to dissipation: an “inverse resistivity”
$|\chi|^{2}/\operatorname{Im}\chi$ (Miller  _et al._ (2016)), and a material
quality factor proportional to
$\left[\partial(\operatorname{Re}\chi)/\partial\omega\right]/\operatorname{Im}\chi$
(Wang and Shen (2006)). By contrast, dielectric materials (Sec. III) require
patterning at sizes at the scale of wavelength Krasnok  _et al._ (2012); Fu
_et al._ (2013); Kuznetsov  _et al._ (2016), and radiative coupling (e.g.
surface roughness) is typically the dominant loss mechanism. For these
materials, a sum rule (generalized from Ref. Barnett and Loudon (1996))
dictates that the all-frequency response is constrained by the real part of
the refractive index, $n$, which becomes a key metric of interest. In Sec. IV,
we review two known ways to compare _between_ the two material categories: a
sum rule that equates total scattering response to the total number of
electrons in the scatterer Gordon (1963); Yang  _et al._ (2015a), and recent
“power-bandwidth” limits that enable comparison over any bandwidth from 0 to
$\infty$ Shim  _et al._ (2019a). We include comparisons between the two and
show that it _is_ possible in many scenarios to quantitatively determine
whether polaritonic or dielectric approaches are optimal. Finally, looking
forward, we examine new avenues of exploration at the intersection of optical-
material synthesis and maximum nanophotonic response (Sec. V).

Given the excitement over recent material breakthroughs, there are a few
recent reviews surveying various aspects of the field Naik  _et al._ (2013);
Caldwell  _et al._ (2015); Zhong  _et al._ (2015); Kuznetsov  _et al._ (2016);
Basov  _et al._ (2016); Bhimanapati  _et al._ (2015); Tan  _et al._ (2017);
Low  _et al._ (2017); Foteinopoulou  _et al._ (2019). In this review, we
include and emphasize only those material characteristics that can be shown to
be globally optimal for some application; moreover, we require such optimality
to be _independent_ of the underlying structure or geometry of the system. We
do not include analyses or properties that are only true for, e.g., spheres or
planar surfaces; instead, we show that in fact there is now extensive
theoretical understanding about structure-independent optimal material
properties.

##  II Polaritonic Materials

Polaritonic materials at optical frequencies benefit from the strong coupling
of light to free electrons, but at the cost of significant absorption. To what
extent can absorption losses be avoided, for large scattering or high-$Q$
resonances? In this section we highlight two results that answer this
question: bounds on the largest single-frequency response possible (Sec. II.1)
and bounds on the highest quality factor possible (Sec. II.2), yielding two
metrics ($|\chi|^{2}/\operatorname{Im}\chi$ and
$\left[\partial(\operatorname{Re}\chi)/\partial\omega\right]/\operatorname{Im}\chi$,
respectively, for bulk materials with susceptibilities $\chi$) by which all
polaritonic materials can be compared. We also highlight the important role
the _real_ part of the permittivity plays to determine the feasibility of
achieving high-confinement polaritonic resonances (Sec. II.3). In Sec. II.4 we
highlight general bounds for maximum response in a regime of high-radiative-
efficiency plasmonics, and hybrid dielectric–metal structures that offer a
combination of high efficiency and large response, approaching their
respective bounds. Finally, we discuss the important role _nonlocality_ plays
at small length scales, and known bounds that incorporate the relevant
nonlocal parameters (Sec. II.5).

###  II.1 Maximal Single-frequency Response

It has long been recognized that reducing material “loss” is critical for many
plasmonics applications, yet there are many possible loss rates to choose
from: the imaginary part of the susceptibility, the imaginary part of the
refractive index, the real part of a material’s conductivity, the inverse of
its real resistivity, etc. One way to frame the question is in the context of
scattering problems: given an external excitation, what is the largest
possible response (absorption, scattering, etc.) from a given material? Miller
_et al._ (2016) develops a systematic answer to this question, independent of
particulars of the geometrical patterning. The idea stems from consideration
of the polarization currents $\boldsymbol{\mathbf{P}}$ excited within the
material. The rate at which energy is absorbed is proportional to
$\int_{V}\left(\operatorname{Im}\chi\right)|\boldsymbol{\mathbf{E}}|^{2}=\left(\operatorname{Im}\chi/|\chi|^{2}\right)\int_{V}\left|\boldsymbol{\mathbf{P}}\right|^{2}$,
and is proportional to the square of the magnitude of the polarization
currents. By contrast, the well-known optical theorem Newton (1976); Lytle
_et al._ (2005); Jackson (2007); Bohren and Huffman (2008) dictates that the
total extinction of the incident field, via absorption or scattering, must be
proportional to the imaginary part of a scattering amplitude, which increases
only linearly with the induced polarization currents. Extinction must be
larger than absorption, and yet absorption increases more rapidly with the
magnitude of the polarization currents. Constraining absorption to be smaller
than extinction, which is equivalent to requiring that scattered power be non-
negative, thereby imposes a limit on the largest polarization currents that
can be excited in any particular material. This constraint can then be used to
identify bounds on many optical-response functions. The exact form of the
bound depends on the application, but within each bound there is a “material
figure of merit” $f(\omega)$ that dictates increased possible response based
on the material properties. (A similar analysis Miller  _et al._ (2017a) can
be done for the induced surface currents in 2D materials with local surface
conductivities $\sigma(\omega)$.) For any polaritonic material, the material
metric is given by

$f(\omega)=\begin{dcases}\frac{|\chi(\omega)|^{2}}{\operatorname{Im}{\chi(\omega)}}&\text{3D
/ bulk materials}\\\
Z_{0}\frac{|\sigma(\omega)|^{2}}{\operatorname{Re}\sigma(\omega)}&\text{2D
materials},\end{dcases}$ (1)

where the impedance of free space, $Z_{0}$, is included in the 2D-material FOM
to make it dimensionless. This material metric determines the maximal single-
frequency response for many applications, including absorption, scattering,
and local density of states Miller  _et al._ (2016, 2014, 2017a), cross
density of states Shim  _et al._ (2019a), near-field radiative heat transfer
Miller  _et al._ (2015), high-radiative-efficiency plasmonics Yang  _et al._
(2017), free-electron radiation Yang  _et al._ (2018), Raman scattering Michon
_et al._ (2019), and more. Equation (1) is for scalar, nonmagnetic materials;
more generally, with a tensor susceptibility $\boldsymbol{\mathbf{\chi}}$ or
conductivity $\boldsymbol{\mathbf{\sigma}}$, the material metric becomes
$\left\|\boldsymbol{\mathbf{\chi}}^{\dagger}\left(\operatorname{Im}\boldsymbol{\mathbf{\chi}}\right)^{-1}\boldsymbol{\mathbf{\chi}}\right\|_{2}$
(Miller  _et al._ (2016)) and
$Z_{0}\left\|\boldsymbol{\mathbf{\sigma}}^{\dagger}\left(\operatorname{Im}\boldsymbol{\mathbf{\sigma}}\right)^{-1}\boldsymbol{\mathbf{\sigma}}\right\|_{2}$
(Miller  _et al._ (2017b)), respectively, where $\left\|\cdot\right\|_{2}$
denotes the matrix 2-norm Trefethen and Bau (1997).

Figure 1: (a) Comparison of representative bulk, polaritonic (and/or lossy)
materials via the material figure of merit
${|\chi(\omega)|^{2}}/\operatorname{Im}{\chi(\omega)}$. Conventional metals
(Ag, Au, Al, Cu) Palik (1998) outperform alternative plasmonic materials
(aluminum-doped ZnO (AZO) Naik  _et al._ (2013), Dysprosium-doped CdO (CdO:Dy)
Sachet  _et al._ (2015), SiC Francoeur  _et al._ (2010), TiN Hibbins  _et al._
(1998), ITO Franzen (2008), doped InAs Law  _et al._ (2013), n-type and p-type
Si Basu  _et al._ (2010)) in the visible and infrared (albeit at the expense
of large permittivity real parts). Drude material susceptibilities exhibit
linear increases in material FOM as a function of wavelength, explaining the
linear trend in the figure. (b) Comparison of various 2D materials by material
FOM
$Z_{0}\left\|\boldsymbol{\mathbf{\sigma}}^{\dagger}\left(\operatorname{Im}\boldsymbol{\mathbf{\sigma}}\right)^{-1}\boldsymbol{\mathbf{\sigma}}\right\|_{2}$
(or $Z_{0}{|\sigma(\omega)|^{2}}/{\operatorname{Re}\sigma(\omega)}$ in the
scalar case). Here, we compare: graphene at different Fermi levels Jablan  _et
al._ (2009) (solid black lines) and magnetic-biasing Hanson (2008) (dashed
black line), AA-stacked bilayer graphene Wang  _et al._ (2016) (dark red), hBN
Brar  _et al._ (2014) (green), MoS2 Liu  _et al._ (2014b) (purple), the
anisotropic conductivity components of black phosphorus Low  _et al._ (2017)
(BP, pink and dark purple), and three 2D metals García de Abajo and Manjavacas
(2015), Al (red), Ag (blue), and Au (gold). High-Fermi-level graphene and 2D
silver offer the largest possible responses at infrared and visible
wavelengths, respectively. The inset compares graphene at THz frequencies Rana
(2008); Ju  _et al._ (2011); Low and Avouris (2014) to the topological
insulator Bi2Se3 Di Pietro  _et al._ (2013), which can have a surprisingly
large FOM. For (b), reprinted with permission from Ref. Miller  _et al._
(2017a), American Chemical Society.

Intuitively, the material metric of Eq. (1) makes sense: larger absolute
susceptibilities imply the ability to sustain large currents, while the
imaginary part of the susceptibility must dampen resonant response. Figure
1(a,b) compares different polaritonic materials against the material FOMs
$f(\omega)$ of Eq. (1) for experimentally characterized bulk and 2D materials.
In the figure for bulk materials, across a spectrum ranging from the extreme
UV to the mid-infrared, there is a clear trend for increasing $f(\omega)$ with
wavelength, which can be attributed to Drude-like response in such materials.
For a Drude susceptibility
$\chi(\omega)=-\omega_{p}^{2}/(\omega^{2}+i\gamma\omega)$, the material FOM is
given by $\omega_{p}^{2}/\gamma\omega$, and is therefore exactly proportional
to wavelength. Variations from linear dependence thus represent non-Drude
features in the material susceptibilities. The increasing material FOM with
wavelength may be compensated by frequency-dependent constants in the response
function; for example, the far-field scattering bounds Miller  _et al._ (2016)
multiply the material FOM by a factor $\omega/c$ that exactly compensates a
linear increase with wavelength. As we discuss further in Sec. II.3, the large
values of $|\chi|^{2}/\operatorname{Im}\chi$ for noble metals at infrared
frequencies may represent bounds that are not achievable in practice (due to
infeasible synthesis requirements), in which case polar-dielectric materials,
transparent conducting oxides, and doped semiconductors may all be viable
alternatives. In the case of 2D materials, per Fig. 1(b), one can see that
graphene with a large Fermi level appears ideal at photon energies below
$1\text{\,}\mathrm{e}\mathrm{V}$, while 2D Ag, Al, and Au all perform very
well at higher photon energies. In that figure we take the 2D-material limit
from bulk properties of the metals; intriguingly, ab-initio calculations
suggest that actual single-layer sheets of 2D materials may have significantly
larger material FOM than their infinitely thin bulk counterparts Sundararaman
_et al._ (2018).

The bulk-material figure of merit of Eq. (1) appears in other contexts as
well. In Arbabi  _et al._ (2014), it is shown that
$|\chi|^{2}/\operatorname{Im}\chi$ is the key material metric determining a
geometry-independent fundamental limit to propagation length in plasmonic
waveguides. The essence of that derivation similarly approaches the problem as
one of identifying the maximum possible induced polarization currents. (It has
been suggested Raether (1988); Barnes  _et al._ (2003); Dionne  _et al._
(2005); Barnes (2006); Homola (2006) that the propagation length of a surface
plasmon on a planar interface follows an expression that ultimately is
proportional to
$(\operatorname{Re}\varepsilon)^{2}/\operatorname{Im}\varepsilon$, which is
very similar to the material metric. However, this is derived with a Taylor
expansion that is invalid on resonance, as explained in Appendix E of Ref.
Miller  _et al._ (2016); the correct expression is in fact proportional to the
$\sqrt{\operatorname{Im}\chi}$, which _decreases_ as loss decreases, because
of the concomitant reduction in group velocity. But these arguments only apply
to planar surfaces and patterned surfaces may more closely approach the bound
of Arbabi  _et al._ (2014).) The inverse of the material metric has also been
identified as the fundamental loss quantity to be minimized in metamaterial-
based models Tassin  _et al._ (2012).

Epsilon-near-zero materials Liberal and Engheta (2017); Kinsey  _et al._
(2019) exhibit intriguing phenomena such as distorted channels Silveirinha and
Engheta (2006); Edwards  _et al._ (2008), high-directivity emission Enoch  _et
al._ (2002); Pacheco-Peña  _et al._ (2014), and arbitrarily large phase
velocities Moitra  _et al._ (2013); Li  _et al._ (2015), and offer the
possibility for significant enhancement of nonlinear optical response Capretti
_et al._ (2015); Alam  _et al._ (2016); Caspani  _et al._ (2016). But they do
not offer any particular benefit for large linear response. At the frequency
where the real part of the permittivity crosses zero, the material FOM
simplifies to
$\left[1+(\operatorname{Im}\chi)\right]^{2}/\operatorname{Im}\chi$, which will
tend to be significantly smaller than many of the values in Fig. 1(a), due to
the modest magnitude of $\chi$.

An ideal polaritonic material has a purely real, negative permittivity with
zero loss, in which case the material FOM diverges. A real, negative
permittivity over a nonzero, finite frequency band is compatible with
causality requirements (e.g., the Kramers–Kronig relations), and in theory
ultra-low loss metals may be achievable in materials with artificially large
lattice spacing or in designer organometallic compounds Khurgin and Sun
(2010). The bounds for any physical power-flow quantity cannot themselves
diverge, and must be regularized by other effects in the presence of a
lossless material. In the case of cross-sections _per volume_ , the bounds
_can_ diverge as nonzero response is possible with arbitrarily small volumes.
However, cross-sections themselves cannot diverge as radiative losses must
become dominant as absorptive losses go to zero Gustafsson  _et al._ (2019);
Kuang  _et al._ (2020); Molesky  _et al._ (2020). In the near field, there is
not necessarily any radiative coupling; one form of regularization would be
the breakdown of a local bulk susceptibility at the large wavevectors that are
accessible with very small loss. In either scenario, lower-loss materials
towards such regularizations would represent improved response relative to the
current state-of-the-art. From a microscopic perspective, two fundamental
sources of loss are intrinsic quantum linewidth broadening Kuzyk (2006) as
well as inhomogeneous broadening, but typical metals exhibit higher losses
than required by these sources and thus significant improvements may be
possible Crowell and Kuzyk (2020).

Figure 2: Figure of merit,
$f(\omega)={|\chi(\omega)|^{2}}/{\operatorname{Im}{\chi(\omega)}}$, of
different experimental data Johnson and Christy (1972); Palik (1998); Olmon
_et al._ (2012); Babar and Weaver (2015); McPeak  _et al._ (2015); Yang  _et
al._ (2015b); Yakubovsky  _et al._ (2017); Ciesielski  _et al._ (2017) for (a)
Ag and (b) Au. The metric can vary dramatically even within the same material,
depending on the synthesis techniques and experimental conditions. The names
represent the authors from whom the data was obtained, and all of the curves
are for bulk, thick films except for the dashed black lines, which are for
thin films.

There has been significant effort in mitigating losses in polaritonic
materials Khurgin and Sun (2010); West  _et al._ (2010); Boltasseva and
Atwater (2011); Khurgin and Boltasseva (2012); Naik  _et al._ (2013); Khurgin
(2015). The synthesis techniques for a given material can dramatically affect
the material FOM, as depicted in Fig. 2. High-quality films of Ag and Au can
be deposited using focused-ion-beam lithography or other techniques McPeak
_et al._ (2015). However, it is shown that, up to a threshold, the thickness
of the film will adversely increase the optical loss Yakubovsky  _et al._
(2017); Ciesielski  _et al._ (2017). The dielectric function of bulk silver is
sensitive to environmental conditions and influenced by extrinsic effects,
such as surface, impurity, and grain boundary scattering Yang  _et al._
(2015b). On the contrary, bulk gold is less sensitive to sample morphology and
variation between different bulk measurements Johnson and Christy (1972);
Palik (1998); Babar and Weaver (2015) are mostly likely caused by systematic
errors Olmon  _et al._ (2012). There are other approaches to mitigating loss
as well. One approach is to engineer the free-carrier concentration Hoffman
_et al._ (2007); Blaber  _et al._ (2010); Naik and Boltasseva (2010); Naik
_et al._ (2011, 2012); Kim  _et al._ (2013a); Jongbum Kim  _et al._ (2013);
Kim  _et al._ (2013b); Agrawal  _et al._ (2018). Another is to use gain media
to compensate for loss Bergman and Stockman (2003); Gather  _et al._ (2010);
Stockman (2011). Polar dielectrics supporting surface phonon-polaritons tend
to naturally have lower losses and are good polaritonic media at mid-infrared
frequencies Caldwell  _et al._ (2015); Low  _et al._ (2017); Caldwell  _et
al._ (2019). While all these approaches help reduce loss as measured by the
imaginary part of susceptibility, they do not necessarily translate to an
enhancement in the material FOM. For example, reducing the free-carrier
concentration decreases the magnitude of the real part of susceptibility, in
addition to decreasing its imaginary part, thus resulting in smaller material
FOM. Thus a useful measure of loss is the inverse of the material FOM,
${\operatorname{Im}{\chi(\omega)}}/{|\chi(\omega)|^{2}}$. Under this metric,
the various approaches mentioned above may not be very effective.
Countertuitively, measures that increase the imaginary part of the
susceptibility could help reduce absorptive losses.

Figure 3: (a) Quality factor (using Eq. (2),valid under quasistatic
approximation) for representative materials as a function of real part of
permittivity. For Ag, optical constants were obtained from both Palik Palik
(1998) and Johnson and Christy (J$\&$C) Johnson and Christy (1972). (b)
Comparison of quality factor (Eq. (2)) for different materials, ranging from
polaritonic materials—metals, metal alloys, and doped semiconductors—to polar
dielectrics. Reprinted with permission from Ref. Caldwell  _et al._ (2015), De
Gruyter.

###  II.2 Quality factor

Quality factor is another measure of loss, indicative of both the relative
loss rate per cycle in a resonator as well as the linewidth of the scattering
contribution from a single resonance. In general, it is a function of both the
resonator geometry and its material properties, but for high-confinement,
highly subwavelength polaritonic resonances, the quasistatic nature of the
resonances implies that material loss is the dominant source of loss and is
geometry independent.

Using simple integral relations for quasistatic fields, Wang and Shen (2006)
derived the quality factor of any low-loss polaritonic resonator:

$\displaystyle
Q=\frac{\omega}{2}\frac{\frac{\partial}{\partial{\omega}}{\left[\operatorname{Re}{\chi(\omega)}\right]}}{{\operatorname{Im}{\chi(\omega)}}}.$
(2)

For Drude materials with loss rate $\gamma$, the $Q$-factor expression
simplifies to $Q=\omega/\gamma$. On physical grounds, Khurgin Khurgin and
Boltasseva (2012) has pointed out that quasistatic resonances have vanishingly
small magnetic-field energies relative to their electric-field energies. The
magnetic-field energy serves the role of “kinetic energy” for the resonator,
and is replaced by the kinetic inductance of the free electrons. Energy stored
in the free electrons, however, will dissipate at a rate proportional to
$\gamma$, independent of the geometrical details of the structure.

Beyond the low-loss regime, Raman  _et al._ (2013) extends the result of Eq.
(2) to any loss regime for Drude-Lorentz-oscillator material models. For
Drude-Lorentz oscillators with damping rates $\Gamma_{n}$, they define a
variable $\gamma_{\rm max}(\omega)$ that is a frequency-dependent weighted
average of $1/2$ times the $\Gamma_{n}$ rates, $\gamma_{\rm
max}(\omega)=\sum_{n}\theta_{n}(\omega)\Gamma_{n}/2$, where $\theta$ is a
spectral weighting factor Raman  _et al._ (2013). Then the quality factor is
bounded below by a simple ratio of the resonant frequency $\omega$ to
$\gamma_{\rm max}$:

$\displaystyle Q\geq\frac{\omega}{2\gamma_{\textrm{max}}(\omega)},$ (3)

where the extra factor of 2 in the denominator arises from the definition of
$\gamma_{\rm max}$ as a weighted average of _half_ of the Drude loss rates.
Technically, Eq. (3) does not require a quasistatic approximation, but it does
require material losses to dominate relative to radiative losses.

These derivations can be adapted easily to 2D materials, by making the
replacement $\omega\chi(\omega)\rightarrow
i\delta_{S}(\mathbf{x})\sigma(\omega)$, where $\delta_{S}(\mathbf{x})$ is a
delta function on the surface of the (not necessarily planar Asmar and Ulloa
(2015)) 2D material. The analog of Eq. (2) for a 2D material is:

$\displaystyle
Q=-\frac{\omega^{2}}{2}\frac{\frac{\partial}{\partial{\omega}}\left[\frac{{\operatorname{Im}{\sigma(\omega)}}}{\omega}\right]}{{\operatorname{Re}{\sigma(\omega)}}}.$
(4)

Beyond the quasistatic approximation, it _is_ possible to narrow the linewidth
beyond the expressions of Eqs. (2)–(4), through e.g. Fano resonances Fano
(1961); Limonov  _et al._ (2017) and near-field coupling Stockman  _et al._
(1996); Kottmann and Martin (2001), but such effects must necessarily occur at
larger size scales, without the highly subwavelength confinement available in
the quasistatic limit.

Figure 3 plots the quality factor computed by Eq. (2) for a wide variety
materials. To compare loss rates at very different frequencies, in Fig. 3(a)
the $Q$ factor of representative materials is shown over a range of moderate
values of the real parts of their permittivities. Figure 3(b) shows the $Q$
factors of many materials at the frequencies for which
$\operatorname{Re}\varepsilon=-2$, where a subwavelength sphere exhibits a
surface-plasmon or surface-phonon resonance. The $Q$ factor near these
resonances can be expressed as $\omega/\gamma$ where $\omega$ refers to the
resonance frequency. The $Q$ factor tends to increase with wavelength because
optical phonon lifetimes for polar dielectrics are typically orders of
magnitude larger than those of plasmonic metals, which more than compensates
for the (roughly an order-of-magnitude) reduction in resonance frequency.
Doped semiconductors are intermediate between the two classes, with their loss
rates several times smaller, but resonance frequencies slightly smaller, than
their metal counterparts.

###  II.3 Real part of permittivity

Polaritonics in the mid-IR spectrum is rich with applications in sensing and
selective thermal emission Greffet  _et al._ (2002); Mason  _et al._ (2011);
Liu  _et al._ (2011), given that a wide variety of molecules exhibit
fundamental vibrational and rotational modes in the mid-IR and that blackbody
emission peaks in this range for typical temperatures Law  _et al._ (2012);
Caldwell  _et al._ (2015); Zhong  _et al._ (2015). As discussed above, metals
can have very large material FOM $f(\omega)=|\chi|^{2}/\operatorname{Im}\chi$
at such frequencies because the real parts of their permittivities tend to
scale as $1/\omega^{2}$. However, achieving the corresponding scattering
bounds may be unrealistic. Achieving polaritonic resonances in materials with
large negative real permittivities may require difficult-to-fabricate feature
sizes. As an example, for ellipsoids to have a quasistatic /plasmonic
resonance at a particular frequency requires their depolarization factors $L$
to equal $\operatorname{Re}(-1/\chi)$ (Miller  _et al._ (2016)), which
requires increasingly large aspect ratios as
$L=\operatorname{Re}(-1/\chi)\rightarrow 0$. As an example, for a resonance at
$5\text{\,}\mu\mathrm{m}$ wavelength in silver would require an aspect ratio
greater than 50, whereas the highest aspect ratios fabricated to date are
roughly 30 (Busbee  _et al._ (2003)). It is possible to shift resonances to
longer wavelengths without high aspect ratios by increasing their size, and
correspondingly the radiative damping, but the confinement is reduced and the
materials start behaving more like perfect conductors rather than plasmonic
materials Law  _et al._ (2013). Large, negative real part of permittivities
are also undesirable for certain transformation-optics based devices and
applications. For instance, Ref. Cai  _et al._ (2007) has designed a non-
magnetic, cylindrical cloak at optical frequencies that require the real part
of permittivity of the constituent metal wires to have a similar magnitude to
that of the surrounding dielectric.

Figure 4: Comparison of (negative) $\operatorname{Re}\varepsilon$ of various
polaritonic materials for a wide range of frequencies, ranging from visible to
mid-IR. The pink shaded region shows the allowed range of
$\operatorname{Re}\varepsilon$ for realistic aspect ratio of 1 to 5 for a
prolate spheroid. Given this range, conventional metals are ideal in the
visible, whereas alternative plasmonic materials are suited for infrared
spectrum. The extent to which the real part of permittivity of different
materials responds to wavelength varies, with SiC demonstrating very small
wavelength range over the pink shaded region and doped InAs the largest in the
mid-IR.

Figure 4 compares the magnitude of $\operatorname{Re}\varepsilon$ for various
polaritonic materials in the visible and infrared spectrum. The shaded region
corresponds to permittivity values whose real parts are between -2 and -16.9,
which for nanorod-like ellipsoidal nanoparticles maps to aspect ratios from 1
to 5. This may be considered the region within which true quasistatic
resonances can be achieved.

A subtlety that arises for high-loss materials is that the negativity of the
real part of the permittivity no longer becomes the true delineation of
whether a material supports quasistatic polaritonic resonances. In quasistatic
electromagnetism, Maxwell’s equations simplify to Poisson’s equation for the
quasistatic fields. Poisson’s equation can be transformed to a surface-
integral equation Fuchs (1975); Ouyang and Isaacson (1989); García de Abajo
and Howie (2002); Mayergoyz  _et al._ (2005); Kellogg (1929); Miller  _et al._
(2014) for surface-charge configurations $\sigma$ at all material interfaces,
a surface-integral equation that can be written in the form
$\hat{K}\sigma-\Lambda\sigma=-s$, where $\hat{K}$ is the Neumann-Poincare
operator Mayergoyz  _et al._ (2005); Kellogg (1929); Sandu (2013) that is a
Green’s-function convolution operator, $\Lambda$ depends on the material
susceptibility $\chi$ (assuming vacuum exterior, easily generalizable) via
$\Lambda=1/2+1/\chi$, and $s$ is a source term proportional to the incident
field. The key aspect relevant to this discussion is that in an
eigendecomposition of $\hat{K}$, the response will be maximal if the real part
of $-1/\chi$ is in the range $[0,1]$. This is the true condition for
plasmonic-like response for a lossy material:
$\operatorname{Re}(-1/\chi)\in[0,1]$, generalizing the simple negative-
permittivity condition for materials with nontrivial loss rates.

###  II.4 High-radiative-efficiency plasmonics

In the preceding sections, it was emphasized that one wants quasistatic
resonances for maximum confinement in polaritonic response, and that coupling
to radiative channels reduces such response. Yet for applications where high
radiative efficiency is important, in areas such as far-field imaging Tam  _et
al._ (2007); Anker  _et al._ (2008), photovoltaics Atwater and Polman (2010);
Brongersma  _et al._ (2014a), and quantum nanophotonics De Leon  _et al._
(2012), it is important to identify bounds that incorporate radiative-
efficiency constraints. This problem is considered in Yang  _et al._ (2017),
where it is shown that for both far-field scattering and near-field LDOS
quantities, one can derive bounds with the additional constraint of high
radiative efficiencies. In particular, the results of Yang  _et al._ (2017)
show that imposing a minimum radiative efficiency $\eta_{\rm min}$ (for hard-
to-achieve radiative efficiencies above 50%) effectively reduces the maximum
possible scattering response by a factor $\eta_{\rm min}\left(1-\eta_{\rm
min}\right)/4$, in which case one can define the material FOM by

$\displaystyle f(\omega)=\frac{\eta_{\rm min}\left(1-\eta_{\rm
min}\right)}{4}\frac{|\chi(\omega)|^{2}}{\operatorname{Im}\chi(\omega)}.$ (5)

The new material FOM, given by Eq. (5), explicitly identifies the tradeoff in
response that must be sacrificed to achieve high radiative efficiency.

Figure 5: Schematic of hybrid dielectric–metal resonator, as shown in the top
left. Scattering and absorption cross sections of (a) Si cylinder in free-
space and (b,c) Si and Ag cylinders, respectively, above a semi-infinite Ag
substrate with gap thickness $g=$ 2 nm. Geometrical parameters (insets) are
chosen to align their resonant wavelengths at 700 nm. The three structures are
all illuminated by normally incident plane waves. In (b,c), the absorption
includes the dissipation in both the particle and the substrate. (d) The
dielectric–metal structure shows the highest per-volume scattering cross-
section, because it simultaneously achieves large scattering cross-section
$\sigma_{\textrm{sca}}$, high radiative efficiency $\eta$, and small particle
volume $V$. (e) In the visible regime, the scattering capabilities of
metal–metal geometries (Ag–Ag and Au–Au bounds), free-space metallic (Ag
bound), and free-space dielectric (Si free-space) scatterers all fall short
when compared with the dielectric–metal (Si–Ag) scatterer, which also
approaches its own upper bound. For the Si–Ag and Ag–Ag structures, the gap
size is fixed at 5 nm; the cylinder (both Si and Ag) height $h$ ranges from 40
to 60 nm in order to tune the resonant wavelength. Reprinted with permission
from Ref. Yang  _et al._ (2017), American Chemical Society.

The radiative-efficiency-constrained bounds enable comparison Yang  _et al._
(2017) of all-metal Esteban  _et al._ (2010); Moreau  _et al._ (2012); Belacel
_et al._ (2013); Rose  _et al._ (2014); Faggiani  _et al._ (2015) and hybrid
metal–dielectric Yang  _et al._ (2017); Devilez  _et al._ (2010); Rusak  _et
al._ (2014) approaches to high-radiative-efficiency plasmonics. Shown in Fig.
5 is one set of findings from Yang  _et al._ (2017): a hybrid silicon-on-
silver resonator could have superior plasmonic properties to an all-silver
resonator when radiative-efficiency constraints are included. Shown in Fig.
5(a-d) are the absorption and scattering cross-sections of silicon-only,
silicon–silver, and silver-only optimized resonator designs, with the hybrid
silicon-on-silver resonator showing the largest scattering cross-sections per
volume. Moreover, as shown in Fig. 5(e), these resonators approach their
scattering-efficiency bounds, including the metric of Eq. (5). Also shown in
Fig. 5(e) are the bounds for silver-only and gold-only nanoresonators, which
lie below the actual scattering performance of the designed silicon-on-silver
structure (solid blue lines). This implies that there is no silver-only or
gold-only approach that can ever do better than the designed silicon-on-silver
structure, no matter how optimized the patterning is. This shows the power of
such bounds: they convey the ability to survey a research field and rank-order
certain approaches relative to each other. As shown in Yang  _et al._ (2017),
the hybrid structures are also superior to metal-only structures for near-
field spontaneous-emission enhancements at high radiative efficiency.

###  II.5 Nonlocal effects

Another important consideration is the effect of _nonlocality_ in material
susceptibilities for media synthesized at single-nanometer length scales,
whereby the polarization currents induced at a point $\boldsymbol{\mathbf{x}}$
are related to the electromagnetic fields at another point
$\boldsymbol{\mathbf{x}}^{\prime}$. Such effects both shift resonant
frequencies García de Abajo (2008); Ciraci  _et al._ (2012); Mortensen  _et
al._ (2014); Raza  _et al._ (2015) as well as dampen the maximal possible
response Yan  _et al._ (2012); Correas-Serrano  _et al._ (2015); Miller  _et
al._ (2017b). Reference Miller  _et al._ (2017b) considers the maximum
response when the material susceptibility is described in a hydrodynamic
framework, where the currents behave like fluids with a diffusion constant $D$
and convection constant $\beta$ (both real-valued), in which case the current
is the solution of a convection–diffusion equation driven by the electric
fields. In a quasistatic framework that is relevant at the length scales where
such effects are important, one can show Miller  _et al._ (2017b) that the
cross-section bounds depend on a competition of two “rates:” the material FOM
$f(\omega)$, and a second term relating the size of the scatterer to the
“diffusion” length in the material. If one defines a radius $r$ of the
smallest bounding sphere containing the scatterer, and a plasmonic diffusion
length $\ell_{D}=\sqrt{cD/\omega_{p}^{2}}$, for plasma frequency $\omega_{p}$,
the maximum extinction cross-section per area of a 2D-material scatterer is
given by Miller  _et al._ (2017b)

$\displaystyle\frac{\sigma_{\rm
ext}}{A}\leq\left[\left(Z_{0}\frac{\left|\sigma_{\rm
loc}\right|^{2}}{\operatorname{Re}\sigma_{\rm
loc}}\right)^{-1}+\left(\frac{r^{2}}{\ell_{D}^{2}}\right)^{-1}\right]^{-1},$
(6)

where $\sigma_{\rm loc}$ is the local contribution to the conductivity.
Equation (6) shows a dramatic reduction in response at size scales well below
$\ell_{D}$, serving both to illuminate the effects of nonlocality as well as
potentially serving as a means to extract the value of $\ell_{D}$ itself for
any 2D material from experimental measurements. The analog of Eq. (6) for bulk
materials is straightforward, replacing the first term with the material FOM
$f(\omega)$ from Eq. (1) and the second term from a volume integral instead of
a surface integral. An intriguing next step would be to consider the bounds
that are possible in ab-initio material models.

##  III High-index dielectric materials

Dielectric materials, with $\operatorname{Re}\varepsilon>0$, can have very
small material loss rates, and their dominant loss channel is typically
radiation. The positive real part of the permittivity prevents such materials
from supporting quasistatic, highly subwavelength resonances (though very
small mode volumes are possible Robinson  _et al._ (2005); Liang and Johnson
(2013); Hu and Weiss (2016); Choi  _et al._ (2017)), and it elevates the
importance of geometrical patterning in their response. Independent of the
patterning, however, there is an important geometry-independent sum rule
governing their response and identifying high refractive index as a key metric
for dielectric materials.

In a homogeneous medium with refractive index $n$, the density of plane-wave
electromagnetic states is proportional to $n^{3}$ and $\omega^{2}$ at
frequency $\omega$ (Yariv (1989)). If we consider an electric dipole radiating
at a any point in the medium, it will efficiently couple to half of these
states (effectively the half with the same polarization). The power that it
radiates will be directly proportional to the density of states, and is called
the (electric) _local_ density of states, LDOS Novotny and Hecht (2012). The
LDOS is a measure of the relative energy density of modes at a given point in
space relative to the total density of modes Oskooi and Johnson (2013).
Denoting the free-space electric LDOS by $\rho_{0}$, it is given by the
expression Joulain  _et al._ (2003)

$\displaystyle\rho_{0}(\omega)=\frac{n^{3}\omega^{2}}{2\pi^{2}c^{3}}.$ (7)

Now consider a dipolar source in a _structured_ medium of refractive index
$n$. For a point source at some position $\boldsymbol{\mathbf{x}}$, the LDOS
$\rho(\boldsymbol{\mathbf{x}},\omega)$ is highly dependent on the position
$\boldsymbol{\mathbf{x}}$, the frequency $\omega$, and the material and
structuring. But, if one considers a certain integral over all frequencies
(and discard a near-field coupling term that typically, though not always,
corresponds to emission into material-absorption pathways), Barnett and Loudon
showed Barnett and Loudon (1996) that there is a simplifying sum rule from
causality arguments. They considered a source point in vacuum, with a
structured medium surrounding it, and showed that the integral over all
frequencies of the relative difference between the structured-medium LDOS and
the vacuum LDOS, i.e. $\left(\rho-\rho_{\rm vac}\right)/\rho_{\rm vac}$, is
precisely 0. We can generalize their argument slightly, pointing out that if
the source is in a medium with constant refractive index $n$ (over all
frequencies), the same sum rule should apply to the structured-medium LDOS
relative to the _background_ LDOS $\rho_{0}$ for a homogeneous medium of
refractive index $n$, as given in Eq. (7). Then the generalization of the sum
rule from Barnett and Loudon (1996) is

$\displaystyle\int_{0}^{\infty}\frac{\rho(\boldsymbol{\mathbf{x}},\omega)-\rho_{0}(\omega)}{\rho_{0}(\omega)}\,{\rm
d}\omega=0.$ (8)

The sum rule of Eq. (8) states that for any patterning, the total LDOS of a
system cannot be modified, on average, over all frequencies. The LDOS can be
increased in one frequency range, but thereby must be reduced in another. The
only way to increase LDOS over all frequencies is by increasing the refractive
index $n$ of the medium itself, which increases $\rho_{0}(\omega)$ by $n^{3}$
per Eq. (7). This highlights the key role that refractive index can play in
maximizing light–matter interactions.

The role of refractive index to increase the available states has been
recognized in a wide variety of systems. Large refractive index enables
smaller mode volumes Robinson  _et al._ (2005); Liang and Johnson (2013); Hu
and Weiss (2016); Choi  _et al._ (2017) and correspondingly large spontaneous-
emission enhancements Purcell (1995); Agio and Cano (2013); Sauvan  _et al._
(2013). In a photonic-crystal cavity, a typical defect mode has minimum mode
volume $\sim\left(\lambda/2n\right)^{3}$, in which case larger refractive
index widens the bandgap and increases the spatial confinement of the mode
Yablonovitch (1993); Foresi  _et al._ (1997); Boroditsky  _et al._ (1998). In
the quest for miniaturization of nanophotonic “building blocks,” plasmonic
split-ring resonators tend to exhibit large absorption and can be difficult to
fabricate and miniaturize at optical frequencies Enkrich  _et al._ (2005);
Soukoulis  _et al._ (2007). High-index dielectric nanoresonators exhibit
strong electric and magnetic resonances Zhao  _et al._ (2009) and can help
overcome such problems Kuznetsov  _et al._ (2012); Evlyukhin  _et al._ (2012).
Large refractive index contrast between the core and air helps to confine the
light within the high index medium. As a result, the reduced radiation loss
improves the overall $Q$ factors of the resonator Kuznetsov  _et al._ (2016).
High refractive index is also necessary for the miniaturization of such
nanoresonators Baranov  _et al._ (2017), since the magnetic resonance of a
sphere occurs at a wavelength $\lambda\approx nd$, where $n$ is the refractive
index and $d$ is the diameter of a sphere Kuznetsov  _et al._ (2012);
Evlyukhin  _et al._ (2012); Staude  _et al._ (2013). With nanoresonators as
building blocks, a high-index material with positive permittivity and unit
permeability can create metamaterials with effective permittivities and
permeabilities across the four possible quadrants Wu  _et al._ (2014); Yang
_et al._ (2014); Jahani and Jacob (2016). Nanoresonators with electric- and
magnetic-dipole resonances can be used to tailor scattering profiles Schuller
_et al._ (2007); Ginn  _et al._ (2012); Geffrin  _et al._ (2012); Fu  _et al._
(2013); Person  _et al._ (2013); Bakker  _et al._ (2015); Vaskin  _et al._
(2018); Komar  _et al._ (2018); as one example, effective negative index can
be used for Huygens’ metasurfaces with no reflection loss and tailored forward
scattering Decker  _et al._ (2015); Liu and Choi (2018). Alternatively, with
only a single dipolar resonance, one can create perfect electric reflectors
with near-unity reflection due to the negligible loss of the dielectric
material Moitra  _et al._ (2014, 2015). Finally, magnetic reflectors feature
maximum electric field at the interface, dramatically enhancing light–matter
interaction on that plane Liu  _et al._ (2014c); Esfandyarpour  _et al._
(2014).

A classical application of the density-of-states enhancement associated with
high index is for absorption enhancement in solar cells. Random surface
texturing on thick films with plane-wave-like states enables full occupation
of all of the internal states, an $n^{3}$ enhancement relative to external
illumination from all angles. The intensity is enhanced in proportion to the
product of the density of states with the wave speed, $c/n$, ultimately
yielding the $4n^{2}$ Yablonovitch limit to absorption enhancement
Yablonovitch (1982); Goetzberger (1981). Wavelength-scale pattering has
enabled much thinner structures to approach, though generally not surpass, the
$4n^{2}$ limit Garnett and Yang (2010); Yu  _et al._ (2010); Han and Chen
(2010); Wang  _et al._ (2012b); Callahan  _et al._ (2012); Martins  _et al._
(2013); Ingenito  _et al._ (2014); Ganapati  _et al._ (2014); Brongersma  _et
al._ (2014b).

Figure 6: Comparison of common high-index materials over transparent frequency
ranges in the visible and IR spectrum. Curves with blue colors represent
materials transparent over the visible and near-IR spectrum (MgF2 Dodge
(1984), SiO2 Malitson (1965); Tan (1998), Al2O3 Dodge (1986), Si3N4 Luke  _et
al._ (2015), ZrO2 Wood and Nassau (1982), LiNbO3 Zelmon  _et al._ (1997), GaN
Barker and Ilegems (1973), ZnS Debenham (1984); Klein (1986), TiO2 DeVore
(1951), ZnSe Marple (1964)), and those with red colors over the mid-IR (Si
Chandler-Horowitz and Amirtharaj (2005), ZnTe Li (1984), GaAs Skauli  _et al._
(2003), Ge Burnett  _et al._ (2016), InP Pettit and Turner (1965); Pikhtin and
Yas’kov (1978); Bass (2010), InAs Lorimor and Spitzer (1965), PbS Zemel  _et
al._ (1965); Bass (2010), Te Bhar (1976), PbSe Zemel  _et al._ (1965), PbTe
Weiting and Yixun (1990)). The chromatic dispersion for the two classes of
materials are measured at wavelengths of 500 nm and 5 $\mu$m respectively,
with darker shades corresponding to higher values of chromatic dispersion.
Figure 7: Dispersion characteristics of high-index materials transparent over
the visible and infrared spectrum, as measured by the chromatic dispersion
${\rm d}n/{\rm d}\lambda$ over representative visible and infrared wavelength
of 500 nm and 5 $\mu$m respectively. Materials with higher refractive indices
tend to be more dispersive.

Figures 6 and 7 show the refractive indices of many common high-index
materials over transparency windows at visible and infrared frequencies. Many
of the materials that are transparent in the visible and near-IR are polar
dielectrics that support phonon polaritons, and thus are not transparent in
the mid-infrared. Conversely, many (though not all) of the materials
transparent in the mid-IR are not transparent in the visible/near-IR. One can
see in both figures that the mid-IR-transparent materials tend to exhibit
larger values of $n$ relative to visible/near-IR-transparent materials,
consistent with various models showing that the refractive index for
semiconductors tends to increase with smaller energy gaps Moss (1985); Hervé
and Vandamme (1994); Ravindra  _et al._ (2007); Tripathy (2015); Baranov  _et
al._ (2017). One tradeoff that tends to come with higher refractive index is
increased chromatic dispersion, as measured by the derivative of the
refractive index with respect to wavelength, i.e., $dn/d\lambda$. Figure 7
clearly shows a nearly linear relationship between refractive index and
chromatic dispersion. Thus an important materials-synthesis question is
whether higher-index materials with small chromatic dispersion can be
synthesized. Such materials would immediately improve the performance of a
wide variety of dielectric metasurfaces Yu and Capasso (2014); Aieta  _et al._
(2015); Khorasaninejad  _et al._ (2015, 2016); Arbabi  _et al._ (2016); Kamali
_et al._ (2018); Phan  _et al._ (2019); Lin  _et al._ (2019); Shim  _et al._
(2019b); Chung and Miller (2019, 2020).

##  IV Comparing Polaritonic and Dielectric Materials

The metrics reviewed in Secs. (II,III) are not compatible with each other,
preventing straightforward comparisons between polaritonic and dielectric
media. The bounds in Sec. II are inversely proportional to the loss rate of
the material, which is practically 0 for many dielectric media. Conversely,
the refractive-index sum rule of Sec. III does not account for the surface
waves that are so important in polaritonic media. In this section, we
highlight two measures that enable direct comparison of the two systems: total
electron number for all-frequency response (Sec. IV.1), and recently developed
power–bandwidth metrics for _any_ bandwidth of interest (Sec. IV.2).

###  IV.1 Electron number

The total number of electrons $N_{e}$ in a system is a key property to
describe the maximum response of any material. Causality requires any linear
electromagnetic response function to be analytic in the upper half of the
complex-frequency plane Landau and Lifshitz (1960), which enables contour-
integral-based “sum rules” to connect response averaged over all frequencies
to certain constants of the scatterer. In quantum systems, this leads to the
well-known Thomas–Reiche–Kuhn sum rule (or “f-sum” rule) Kuhn (1925); Thomas
(1925); Levinger  _et al._ (1957); Baxter (1992); Wang (1999), which relates
the sum of oscillator strengths for energy level transitions to the electron
number. One can apply this technique to the extinction cross-section of any
optical scatterer, $\sigma_{\rm ext}(\omega)$, yielding the sum rule Gordon
(1963); Yang  _et al._ (2015a):

$\displaystyle\int_{0}^{\infty}\sigma_{\rm ext}(\omega)\,{\rm
d}\omega=\frac{\pi\omega_{p}^{2}}{2c}V=\frac{\pi
e^{2}}{2\varepsilon_{0}m_{e}c}N_{e}\approx 1.67\times
10^{-5}N_{e}\,[\textrm{m}^{2}\textrm{s}^{-1}],$ (9)

where $\omega_{p}$ is the effective plasma frequency of the material and $V$
is the volume it occupies. Equation (9) says that the extinction of light
summed over all frequencies is determined by the number of electrons in the
scatterer, independent of its geometry and the incident-field polarization. In
Ref. Yang  _et al._ (2015a), extinction cross-sections are computed for
different canonical geometries and materials including aluminum and silicon,
verifying that the sum rule is indeed independent of the nanostructure and
material platform. To maximize frequency-integrated extinction, it is
advantageous to use materials with large electron number. However, Eq. (9)
does not provide any guarantees about which frequency ranges will contain the
resonances. The sum rule requires susceptibilities that satisfy Kramers–Kronig
relations, diminishing to zero at high frequencies. The decay-to-zero
requirement, though physically reasonable, means that even “dielectric” media
(semiconductors, insulators, etc.) have a plasma-like response at large enough
frequencies. Such response contributes to integrated extinction, often in a
large way due to the negative susceptibility. This obscures the behavior of,
for example, a transparent dielectric at optical frequencies, by accounting
for transitions that occur at UV and X-ray frequencies and which can be the
dominant contributor to the response of Eq. (9).

###  IV.2 Maximal response over nonzero, finite bandwidth

Figure 8: (a,d,g) Isocurves of material FOM for a Drude metal (with material
loss rate $\gamma=0.1\omega_{p}$), a lossless dielectric (of susceptibility
$\chi=9$), and a Drude 2D material (with $\gamma=0.01\omega_{p}$). The arrows
indicate increasing material FOM in each case. (b) Comparison of material FOM
for various bulk metals and polaritonic materials (those supporting surface-
phonon polaritons) / dielectrics, keeping the bandwidth-to-center-frequency
ratio $\Delta\omega/\omega_{0}$ fixed to 0.1. Part (c) compares surface-
phonon-polariton-supporting materials at mid-IR wavelengths. (e),(f)
Comparison of material FOM for varying bandwidths relative to the center
wavelengths of 1.55 and 10 $\mu$m. At very narrow bandwidths, dielectrics
offer greater possible response than metals. (h,i) Comparison of material FOM
for 2D materials for different choices of center wavelength and
$\Delta\omega/\omega_{0}$. (2D Al, Ag, and Au properties derived from their
bulk counterparts.) Reprinted with permission from Ref. Shim  _et al._
(2019a), APS.

While the Thomas–Reiche–Kuhn sum rule provides information on integrated
optical cross section, it does not provide spectral information over any
finite bandwidth and often overestimates dielectric-material interactions as
described above—the sum rule weighs the entire spectrum equally instead of
isolating particular frequencies of interest. To remedy these shortcomings,
Ref. Shim  _et al._ (2019a) establishes power–bandwidth limits by combining
causality principles, which underlie the sum rules discussed in Sec. III and
Sec. IV.1, with the energy-conservation principles underlying the single-
frequency bounds of Sec. II. The resulting bounds enable comparisons between
polaritonic and dielectric media, and yield the single-frequency and all-
frequency bounds in their respective asymptotic limits. The key is to connect
the frequency-averaged response over a bandwidth $\Delta\omega$ around a
center frequency $\omega_{0}$ to a single _complex-valued_ frequency
$\omega=\omega_{0}+i\Delta\omega$ (Shim  _et al._ (2019a); Hashemi  _et al._
(2012); Liang and Johnson (2013)). (This simple form emerges for a Lorentzian
average; other averaging “windows” can be used Shim  _et al._ (2019a); Liang
and Johnson (2013).) Then one can derive bounds that depend on material
figures of merit for which the material parameters $\chi$ or $\sigma$ are
evaluated at this complex frequency, i.e. $\chi(\omega)$ or $\sigma(\omega)$.
Then, for nonmagnetic materials with bulk susceptibilities $\chi(\omega)$ or
2D conductivities $\sigma(\omega)$, the material FOM is given by:

$f(\omega=\omega_{0}+i\Delta\omega)=\begin{dcases}\frac{\left|\omega\chi\right|^{2}+\left|\omega\chi\right|\Delta\omega}{\left|\omega\right|\operatorname{Im}\left(\omega\varepsilon\right)}&\text{3D
/ bulk materials}\\\
\frac{|\sigma(\omega)|^{2}}{\operatorname{Re}\sigma(\omega)}&\text{2D
materials}.\end{dcases}$ (10)

(For the general case of anisotropic, magnetic, and even spatially
inhomogeneous media, see the Supplemental Material of Ref. Shim  _et al._
(2019a).) It is evident from Eq. (10) that the material FOM over any nonzero
bandwidth yields a finite value even for lossless materials (due to the
imaginary part of $\omega$, given by $\operatorname{Im}\omega=\Delta\omega$),
and thus enables comparison among all possible optical materials. The FOM for
2D materials has an identical functional form (albeit with a complex
frequency) as the single-frequency 2D-material FOM, Eq. (1), whereas the
functional form of the bulk-material FOM is now slightly more complex in Eq.
(10) than its single-frequency counterpart in Eq. (1). As in the single-
frequency case, the material FOM in Eq. (10) favors large $|\chi(\omega)|$ and
small $\operatorname{Im}{\chi(\omega)}$ (and similarly for 2D materials), now
evaluated at the complex frequency $\omega=\omega_{0}+i\Delta\omega$. One can
identify intuitive forms of the material FOMs for small bandwidth
($\Delta\omega\ll\omega_{0}$), for which the material FOM simplifies to:

$f(\omega)\approx\begin{dcases}\frac{|\chi(\omega)|^{2}}{\operatorname{Im}{\chi(\omega)}}&\text{lossy
(e.g. polaritonic)}\\\
\frac{\omega_{0}}{\Delta\omega}\frac{\left[\chi(\omega)\right]^{2}}{\chi(\omega)+1}&\text{lossless
(dielectric)}\\\
\frac{|\sigma(\omega)|^{2}}{\operatorname{Re}\sigma(\omega)}&\text{2D
materials},\end{dcases}$ (11)

where the 2D material FOM retains its original simple form. For high-index
lossless materials:

$f(\omega)\approx\frac{\omega_{0}}{\Delta\omega}\chi(\omega)=\frac{\omega_{0}}{\Delta\omega}\left(n^{2}-1\right)\qquad\text{lossless,
high-index}.\\\ $ (12)

The material FOM for metals in the small-bandwidth limit is dictated by
material loss $\operatorname{Im}{\chi(\omega)}$, whereas the FOM for lossless
materials (dielectrics) is dictated by relative bandwidth
$\Delta\omega/\omega_{0}$. Lossy materials simplify to the single-frequency
metric $|\chi|^{2}/\operatorname{Im}\chi$ because the bandwidth of their
resonant response is dictated by material loss, and their single-frequency
response can be maintained over the whole bandwidth $\Delta\omega$. The
material FOM for lossless high-index materials is consistent with the metrics
of Sec. III, where we saw that all-frequency response is determined by the
refractive index; the material FOM of Eq. (12) increases in proportion to
$\omega_{0}$ over $\Delta\omega$, which can be interpreted as the possibility
for resonant amplification of the typical $n^{2}$ enhancement in a resonator
with $Q$-factor given by $\omega_{0}/\Delta\omega$. No assumption of single-
mode or quasistatic behavior is made in Eq. (11), which holds for any number
of resonances as well as more complex phenomena such as Fano interactions Fano
(1961); Limonov  _et al._ (2017) and exceptional points Kato (2013); Heiss
(2004).

Figure 8 compares the material FOM for a large variety of materials at optical
frequencies. On the left side of the figure is the material FOM for canonical
material types: (a) a Drude metal,
$\chi(\omega)=-\omega_{p}^{2}/(\omega^{2}+i\gamma\omega)$, for plasma
frequency $\omega_{p}$ and loss rate $\gamma$, (d) a lossless, constant-
susceptibility ($\chi(\omega)=9$) material, and (g) a “Drude” 2D material,
with conductivity $\sigma(\omega)=i\omega_{p}/(\omega+i\gamma)$. One can see
that these three material types show very different characteristic
dependencies of their FOM on frequency and bandwidth. The Drude-metal FOM is
nearly independent of bandwidth for small-to-moderate bandwidths and increases
with the center wavelength (of the frequency band of interest), $\lambda_{0}$.
On the other hand, a dielectric with constant permittivity is independent of
center wavelength, and highly dependent on the bandwidth. Finally, 2D Drude
conductors are somewhere in between. Loss originates from both the material
parameter $\gamma$ as well as the bandwidth, and the material FOM favors small
bandwidth and large wavelength (for a large conductivity). These simplified
permittivity/conductivity models describe the key features of the FOM for real
materials, as the plots in Fig. 8(b,c,h) follow the same trends as those in
Fig. 8(a,d,g): the material FOM for metals increases with wavelength, whereas
dielectrics (Si and SiC) and polaritonic materials (SiO2 and TiO2) that
support surface-phonon polaritons at mid-IR frequencies Maier (2007) are
relatively constant with wavelength. Conversely, the plots in Fig. 8(e,f,i)
show the effects of increasing bandwidth, with the FOM values nearly unchanged
for metals but those of the dielectrics and polaritonic materials decreasing
nearly linearly. The material FOM of 2D conductors increases with both
wavelength and smaller bandwidths.

##  V Looking Forward

The results highlighted in this paper demonstrate a synergy between
experimental materials discovery and theoretical nanophotonic bounds. There
are now a number of key metrics by which materials can be evaluated for
optical performance. Looking forward, these results prompt new questions in a
variety of directions. First, the metrics can drive new-material synthesis,
whether for moderative-negative-permittivity polaritonic material with
particularly small loss (as measured by $\operatorname{Im}\chi/|\chi|^{2}$) or
for dielectric materials with particularly large refractive indices $n$ and
small chromatic dispersion. From a theoretical perspective, a quantum-
mechanical analysis may suggest constituent atoms or alternative approaches to
achieving anomalously large material figures of merit, or they may provide
insight through novel bounds on how large such metrics can be. There is also
the question of which materials are optimal for quantum-photonic
_applications_ , where high radiative efficiency (as discussed in Sec. II.4)
is important, and a number of considerations beyond maximum response may be
desirable. This highlights another area for exploration—alternative
nanophotonic metrics. Beyond maximum response, quantities such as
nonreciprocal transmission, isolation, selectivity, and others may be
desirable Shen  _et al._ (2014); Tzuang  _et al._ (2014); Yu  _et al._ (2015);
Shi  _et al._ (2015); bounds on such response functions may introduce new
material metrics for such scenarios. _Active_ nanophotonic platforms offer
another area for exploration, with metrics such as switching speed taking
increased importance. These examples provide a glimpse at the fertile
opportunity for better understanding of the extreme limits of light–matter
interactions.

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