# Optical response of two-dimensional Dirac materials with a flat band

Chen-Di Han  School of Electrical, Computer and Energy Engineering, Arizona
State University, Tempe, Arizona 85287, USA  Ying-Cheng Lai  Ying-
<EMAIL_ADDRESS>School of Electrical, Computer and Energy Engineering,
Arizona State University, Tempe, Arizona 85287, USA  Department of Physics,
Arizona State University, Tempe, Arizona 85287, USA

###### Abstract

Two-dimensional Dirac materials with a flat band have been demonstrated to
possess a plethora of unusual electronic properties, but the optical
properties of these materials are less studied. Utilizing
$\alpha$-$\mathcal{T}_{3}$ lattice as a prototypical system, where
$0\leq\alpha\leq 1$ is a tunable parameter and a flat band through the conic
intersection of two Dirac cones arises for $\alpha>0$, we investigate the
conductivity of flat-band Dirac material systems analytically and numerically.
Motivated by the fact that the imaginary part of the optical conductivity can
have significant effects on the optical response and is an important factor of
consideration for developing $\alpha$-$\mathcal{T}_{3}$ lattice based optical
devices, we are led to derive a complete conductivity formula with both the
real and imaginary parts. Scrutinizing the formula, we uncover two phenomena.
First, for the value of $\alpha$ in some range, two types of optical
transitions coexist: one between the two Dirac cones and another from the flat
band to a cone, which generate multi-frequency transverse electrical
propagating waves. Second, for $\alpha=1$ so the quasiparticles become
pseudospin-1, the flat-to-cone transition can result in resonant scattering.
These results pave the way to exploiting $\alpha$-$T_{3}$ lattice for optical
device applications in the terahertz frequency domain.

##  I Introduction

Quantum materials whose energy band consists of a pair of Dirac cones and a
topologically flat band, electronic or optical, constitute a frontier area of
research [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28]. For example, in a dielectric photonic
crystal, Dirac cones can be induced by accidental degeneracy occurring at the
center of the Brillouin zone, which effectively makes the crystal a zero-
refractive-index metamaterial at the Dirac point where the Dirac cones
intersect with another flat band [7, 8, 9, 14, 18]. Alternatively, configuring
an array of evanescently coupled optical waveguides into a Lieb lattice [11,
15, 16, 19] can lead to a gapless spectrum consisting of a pair of common
Dirac cones and a perfect flat middle band at the corner of the Brillouin
zone. Loading cold atoms into an optical Lieb lattice provides another
experimental realization of the gapless three-band spectrum at a smaller scale
with greater dynamical controllability of the system parameters [17]. Dice or
$\mathcal{T}_{3}$ optical lattices also possess the Dirac cone and flat-band
structure [29, 30, 2, 5, 10, 31]. Electronically, Dirac materials that can
generate a topologically flat band include transition-metal oxide
SrTiO3/SrIrO3/SrTiO3 trilayer heterostructures [6], $2$D carbon or MoS2
allotropes with a square symmetry [32], as well as SrCu2(BO3)2 [12] and
graphene-In2Te2 bilayer [13].

In two-dimensional (2D) Dirac materials with a flat band, the quasiparticles
are of the massless or massive pseudospin-1 type. Comparing with the
conventional Dirac cone systems with massless pseudospin-1/2 quasiparticles
(e.g., graphene) [33, 34], pseudospin-$1$ systems can exhibit quite unusual
and unconventional physics such as super-Klein tunneling for the two conical,
linearly dispersive bands [3, 35, 5, 18, 36], diffraction-free wave
propagation and novel conical diffraction [11, 15, 16, 19], flat-band rendered
divergent DC conductivity with a tunable short-range disorder [37],
unconventional Anderson localization [38, 39], flat-band ferromagnetism [40,
41, 17], and peculiar topological phases under external gauge fields or spin-
orbit coupling [42, 43, 6, 44]. In particular, topological phases arise due to
the flat band that permits a number of degenerate localized states with a
topological origin, i.e., “caging” of carriers [45]. Additional phenomena
include superscattering of pseudospin-1 waves in the subwavelength regime
[46], geometric valley Hall effect and and valley filtering [47], chaos based
Berry phase detection [48], atomic collapse in pseudospin-1 systems [49],
anomalous chiral edge states in spin-1 Dirac-Weyl quantum dots [50], anomalous
in-gap edge states in two-dimensional pseudospin-1 Dirac-Weyl insulators [51],
and the analogy between Klein scattering of spin-1 Dirac-Weyl wave and
localized surface plasmon [52]. There was also a study of interplay between
classical chaos and flat-band physics [53]. In the past few years, magic-angle
twisted bilayer graphene, a type of quantum materials hosting a flat band, has
become a forefront area of research. These materials can generate surprising
physical phenomena such as superconductivity [54, 55], orbital ferromagnetism
[56, 57], and the Chern insulating behavior with topological edge states.
Notwithstanding the diversity and the broad scales to realize the band
structure that consists of two conical bands and a characteristic flat band
intersecting at a single point in different physical systems, theoretically a
unified framework exists: the generalized Dirac-Weyl equation for massless or
massive spin-1 particles [2, 36].

The optical properties of 2D Dirac materials without a flat band have been
extensively investigated, such as the optical responses of graphene [58, 59,
60, 61], its theoretically predicted frequency-dependent optical conductivity
[62, 63, 64, 65, 66, 67] and experimental verification [68, 69, 70],
suggesting the possibility of developing graphene-based tunable terahertz
optical devices. Such devices can have applications ranging from light
transform [58, 71, 72] and high frequency communication [73, 74, 75] to
cloaking or superscattering [76, 77]. In particular, the discovery of novel
plasmon mode in graphene led to the first experimental superscattering system
at a macroscopic scale [78]. These efforts have given birth to new fields such
as topological photonics [79, 80] and topological lasing [81, 82]. Besides
single-layer graphene, the optical properties of other 2D material have also
been studied. For example, surface plasmon has been uncovered and
characterized in hexagonal boron nitride (hBN) or graphene hBN hybrid
structures [83, 84], in bilayer [65, 85] and multilayer [86, 87] graphene, in
twisted graphene bilayer [88, 89, 90, 91, 92], in silicene [93] and hyperbolic
materials [94, 95]. Particularly worth mentioning is the work on the
conductivity for gapped Dirac fermions [96], where the Kubo formula was used
to calculate the contribution of the intra-band transitions to the optical
conductivity. A central goal in these studies is to garner a strong optical
response at certain frequency or at multiple frequencies. In 2D Dirac
materials without a flat band, transverse magnetic (TM, or $p$-polarized) and
transverse electric (TE, or $s$-polarized) polaritons have been found to
emerge at different frequencies, where the TE polarization can arise at high
frequencies with a low loss [66].

For Dirac materials with a flat band, most previous work focused on their
electronic properties: their optical properties have been less studied. In
this paper, we investigate the “complete” optical responses of 2D Dirac
materials with a flat band, in the sense that both the real and imaginary
parts of the optical conductivity are derived, using the
$\alpha$-$\mathcal{T}_{3}$ lattice as a paradigmatic model system of such
materials. This lattice is formed by adding an atom at the center of each unit
cell of the honeycomb graphene lattice [2], where the low energy excitations
can be described by the pseudospin-1 Dirac-Weyl equation. The parameter
$\alpha$ characterizes the interaction strength between the central atom and
any of its nearest neighbors, relative to that between two neighboring atoms
at the vertices of the hexagonal cell. For $\alpha=0$ there is no coupling
between the central atom and a vertex , so the lattice degenerates to graphene
with pseudospin-1/2 quasiparticles. As the value of $\alpha$ increases from
zero to one, a flat band through the conic interaction of the two Dirac cones
emerges and its physical influences become progressively pronounced [10, 97].
For $\alpha=1$, the lattice generates pseudospin-1 quasiparticles. The flat
band can lead to physical phenomena such as the divergence of conductivity
[37, 98, 49]. Under a continuous approximation, an $\alpha$-$\mathcal{T}_{3}$
lattice can be treated as a thin layer with certain surface conductivity.
Unlike graphene, here the surface conductivity is contributed to by three
types of transitions between the bands: intraband transition, cone-to-cone
transition, and flat-band-to-cone transition. Previously, the optical
conductivity of $\alpha$-$\mathcal{T}_{3}$ lattice was “partially” studied in
the sense that only the real part of the conductivity has been derived [97,
99, 100, 101]. Considering that the imaginary part can affect the optical
response significantly and is therefore important for developing
$\alpha$-$\mathcal{T}_{3}$ lattice based optical devices, we are led to derive
a complete conductivity formula with both the real and imaginary parts. The
formula is verified through two independent approaches and leads to two
previously uncovered phenomena. First, while the intraband transition leads to
TM polarized waves at low frequencies (1-10 THz), TE polarized waves can
emerge at high frequencies (100-300 THz), due to the two interband
transitions. Second, the unique flat-band-to-cone transition generates multi-
frequency TE propagating waves for $\alpha\in(0.4,0.6)$ and a strong optical
response for $\alpha=1$. These phenomena are confirmed through studying the
behaviors of propagating surface wave and scattering.

We remark that a viable experimental way to realize $\alpha$-$\mathcal{T}_{3}$
lattice is through photonic crystals [29, 27, 102, 103], where the three
nonequivalent atoms in a unit cell can be simulated by using coupled
waveguides generated by laser inscription [103]. Electronic materials can also
be exploited to generate pseudospin-1 lattice systems such as transition-metal
oxide $\text{SrTiO}_{3}/\text{SrIrO}_{3}/\text{SrTiO}_{3}$ trilayer
heterostructures [6], $\text{SrCu}_{2}(\text{BO}_{3})_{2}$ [12] or
graphene-$\text{In}_{2}\text{Te}_{2}$ [13]. Recently, materials with a flat
band have been reported [104] A review of the experimentally realized dice-
like system can be found in Ref. [105]. The physics of certain solid-state
materials is effectively that of an $\alpha$-$\mathcal{T}_{3}$ lattice. For
example, a theoretical analysis and computations showed [106] that the
material $\text{Hg}_{1-x}\text{Cd}_{x}\text{Te}$ is equivalent to the
$\alpha$-$\mathcal{T}_{3}$ lattice with $\alpha=1/\sqrt{3}\approx 0.58$. This
material has been realized in experiments [107, 108, 109]. For this reason, we
emphasize the case of $\alpha=1/\sqrt{3}\approx 0.58$ in this paper.

In Sec. II, we describe the optical conductivity for
$\alpha$-$\mathcal{T}_{3}$ lattice, which consists of three parts, and we use
two different methods to derive the conductivity. In Sec. III, we solve the
Maxwell’s equations for $\alpha$-$\mathcal{T}_{3}$ lattice in a dielectric
medium and characterize the properties of the TM and TE polarized waves using
the loss and attenuation length. In Sec. IV, we study optical scattering from
a sphere coated with $\alpha$-$\mathcal{T}_{3}$ lattice and discuss potential
optical device applications. Conclusions and a discussion are offered in Sec.
V.

##  II Optical conductivity of $\alpha$-$\mathcal{T}_{3}$ lattice

Figure 1: Illustration of an $\alpha$-$\mathcal{T}_{3}$ lattice, its band
structure, optical transition, and a possible $\alpha$-$\mathcal{T}_{3}$
lattice based optical devices. (a) An $\alpha$-$\mathcal{T}_{3}$ lattice,
where A, B and C are the three non-equivalent atoms. The hopping energy
between A and B is $t$ (solid line) and that between B and C is $\alpha t$
(dashed line). (b) Band transitions in the $\alpha$-$\mathcal{T}_{3}$ lattice,
where the conduction, valence and flat bands are shown. For a positive
chemical potential at zero temperature, states below it are filled and states
above are empty. The three arrows indicate three band transitions, each
contributing to the optical conductivity. (c) TE mode in an
$\alpha$-$\mathcal{T}_{3}$ lattice placed between two infinite media with
dielectric constants $\epsilon_{1}$ and $\epsilon_{2}$, respectively. (d) A
dielectric sphere coated with multilayer $\alpha$-$\mathcal{T}_{3}$ lattices.
The sphere has dielectric constant $\epsilon_{1}$ and radius $r_{1}$. The
multilayer system has depth $r_{2}$ and $N$ layers of
$\alpha$-$\mathcal{T}_{3}$ lattice with the dielectric function
$\epsilon_{\alpha,N}$.

An $\alpha$-$\mathcal{T}_{3}$ lattice is generated by placing an additional
atom at the center of each unit cell of the honeycomb lattice, where there are
three nonequivalent atoms, as shown in Fig. 1(a). The range of the variation
of the parameter $\alpha$ is $[0,1]$, where the quasiparticles are
pseudospin-1/2 for $\alpha=0$ and pseudospin-1 for $\alpha=1$, and a flat band
arises for $0<\alpha\leq 1$. For convenience, we say that the quasiparticles
for $0<\alpha<1$ are of the “hybrid” type. Under the tight-binding
approximation, the low-energy excitation Hamiltonian is given by [10, 97]

$H_{\alpha}=v_{F}\hbar\begin{pmatrix}0&f_{\mathbf{k}}\cos(\phi)&0\\\
f_{\mathbf{k}}^{*}\cos(\phi)&0&f_{\mathbf{k}}\sin(\phi)\\\
0&f_{\mathbf{k}}^{*}\sin(\phi)&0\end{pmatrix},$ (1)

where $f_{\mathbf{k}}=vk_{x}-ik_{y}$, $\alpha\equiv\tan(\phi)$, $v_{F}$ is the
Fermi velocity, and $v=\pm 1$ denotes the valley index. Solving the eigenvalue
problem associated with the tight-binding Hamiltonian matrix, we obtain three
eigenfunctions, as detailed in Appendix A. There are three distinct energy
bands: a pair of Dirac cones and a flat band, as shown in Fig. 1(b).

The optical properties of the $\alpha$-$\mathcal{T}_{3}$ lattice are largely
controlled by the surface conductivity $\sigma$, which is determined by the
expectation value of the current operator. In the $x$ direction as specified
in Fig. 1(c), the current operator is $j_{x}=-ev_{F}S_{x}$, where

$S_{x}=\begin{pmatrix}0&\cos\phi&0\\\ \cos\phi&0&\sin\phi\\\
0&\sin\phi&0\end{pmatrix}.$ (2)

Expanding the current vector in the three bands, we obtain the matrix
representation of $j_{x}$, as described in Appendix A. For weak electrical
field, the conductivity is given by the standard Kubo formula [63, 67]

$\begin{split}\sigma_{xx}(\omega,\phi)=\frac{\hbar}{2i\pi^{2}}\sum_{n,m}\frac{F(E_{m})-F(E_{n})}{E_{n}-E_{m}}\times\\\
\left(\frac{\langle n|j_{x}|m\rangle\langle
m|j_{x}|n\rangle}{E_{n}-E_{m}-\hbar\omega}+\frac{\langle
m|j_{x}|n\rangle\langle
n|j_{x}|m\rangle}{E_{m}-E_{n}-\hbar\omega}\right),\end{split}$ (3)

where the subscript in $\sigma$ indicates the direction of the current and
electric field, and $\omega$ is the frequency of the electromagnetic wave. For
a homogeneous material without a magnetic field, we have
$\sigma_{xx}=\sigma_{yy}$ and $\sigma_{xy}=\sigma_{yx}=0$. The summation is
over all states $|n\rangle=|\mathbf{k};\pm,0\rangle$ and
$|m\rangle=|\mathbf{k}^{\prime};\pm^{\prime},0^{\prime}\rangle$ with the
respective energy $E_{n}$ and $E_{m}$. Evaluating the integral, the nonzero
terms appear only for [110] $\mathbf{k}=\mathbf{k}^{\prime}$. The quantity $F$
in Eq. (3) stands for the Fermi-Dirac distribution. At zero temperature, the
only transitions allowed are those from the filled to the unfilled bands, or
vice versa.

To obtain the optical conductivity, it is necessary to evaluate the summation
in the Kubo formula Eq. (3). The details of the derivation are presented in
Appendix B.1. The validity of the derivation and the result can be established
by using the Kramers–Kronig (KK) relation as described in Appendix B.2, which
gives the same results. Here we summarize our complete conductivity formulas.

Physically, momentum conservation stipulates that the summation for
$|n\rangle$ and $|m\rangle$ can be regarded as corresponding to band
transition processes. In particular, the summation can be divided into three
parts, denoted as $\sigma^{(1)}(\omega,\phi)$, $\sigma^{(2)}(\omega,\phi)$ and
$\sigma^{(3)}(\omega,\phi)$, which correspond to the intraband, cone-to-cone
and flat-band-to-cone transitions, respectively. The conductivity due to the
intraband transition is

$\sigma^{(1)}(\omega,\phi)=4\mu\sigma_{0}\delta(\hbar\omega)+\frac{4i\mu\sigma_{0}}{\pi\hbar\omega},$
(4)

where $\sigma_{0}=e^{2}/(4\hbar)$ and $\mu$ is the chemical potential. The
Drude peak is represented by a $\delta$-function with the coefficient
proportional to the chemical potential $\mu$. The conductivity due to the
cone-to-cone transition is

$\begin{split}\sigma^{(2)}(\omega,\phi)=&\cos^{2}(2\phi)\sigma_{0}\times\\\
&\left[\Theta(\hbar\omega-2\mu)-\frac{i}{\pi}\ln\left|\frac{\hbar\omega+2\mu}{\hbar\omega-2\mu}\right|\right],\end{split}$
(5)

where $\Theta$ is the Heaviside step function. The conductivity due to the
flat-band-to-cone transition is

$\begin{split}\sigma^{(3)}(\omega,\phi)=&2\sin^{2}(2\phi)\sigma_{0}\times\\\
&\left[\Theta(\hbar\omega-\mu)-\frac{i}{\pi}\ln\left|\frac{\hbar\omega+\mu}{\hbar\omega-\mu}\right|\right].\end{split}$
(6)

For convenience, we introduce a unit free conductivity and divide it into real
and imaginary parts

$\sigma=\sigma^{\prime}+i\sigma^{\prime\prime}=\sigma_{0}(\tilde{\sigma}^{\prime}+i\tilde{\sigma}^{\prime\prime}).$
(7)

At finite temperatures, the Fermi-Dirac distribution is no longer a step
function and this will lead to a change in the conductivity formula, as
detailed in Appendix B.3.

We have also analyzed the effect of finite impurity scattering on the optical
conductivity, as detailed in Appendix B.4. The main result is that finite
impurity scattering will change the conductivity as in Eq. (4) for
$\omega\rightarrow 0$.

For 2D Dirac materials such as graphene, the high frequency regime above
$10$THz is physically important for the optical conductivity. Nonetheless, the
behavior of the conductivity in the low frequency regime can be analyzed in
terms of the product $\omega\sigma^{\prime\prime}$ between the frequency and
the imaginary part of the conductivity (see Appendix B.5). This is motivated
by the fact that, in the study of optical conductivity of superconducting
materials, the quantity $\omega\sigma^{\prime\prime}$ is often used to
characterize the penetration depth [111, 112, 113]. We also study the effects
of varying $\alpha$ on the optical conductivity and find that the flat band
does not play a significant role in the conductivity in the low frequency
regime. However, decreasing the Fermi energy or increasing the temperature can
make the flat-band contribution more pronounced. The different relaxation time
for different values of $\alpha$ can also impact the optical conductivity.

For $\alpha=0$ (or equivalently, $\phi=0$), our conductivity formula reduces
to the one for graphene [62, 63, 64, 65, 66, 67], which was experimentally
verified [68, 69, 70]. For $\alpha\neq 0$, the real part of the conductivity
is consistent with the previous result [97]. The imaginary part of the
conductivity is the contribution of our present work.

Features of the derived conductivity formulas Eqs. (4-6), are as follows.
First, the conductivity does not depend on the detailed lattice structure,
insofar as there are a pair of Dirac cones and a flat band. In fact, the
conductivity is determined by the linear dispersion relationship and the flat
band. Our derivation thus holds for different types of lattices described by
the effective Hamiltonian in (1). Second, the formulas hold only for a
reasonable range in $\omega$ or $\mu$. If the value of $\omega$ is small,
impurity scattering will be important and, in this case, it is necessary to
make the change $\omega\rightarrow\omega+i\tau^{-1}$, where $\tau$ is the
relaxation time. This change will smooth out the $\delta$-function in the
intraband conductivity formula.

For the $\alpha$-$\mathcal{T}_{3}$ model, the value of the relaxation time has
not been available yet, but insights can be gained by considering graphene,
where the experimentally measured [114] relaxation time is about $10^{-13}$s.
Since the optical response of graphene is appreciable at high frequencies
above $10$THz, a relaxation time on the order of $10^{-13}$s means that the
direction of the field changes much faster than that in the scattering caused
by impurity. In this case, impurity scattering can be ignored [115]. If the
value of $\omega$ approaches zero, the conductivity in Eq. (4) will diverge,
which is related to the minimal conductivity problem in graphene that remains
unresolved [116]. If the value of $\omega$ or $\mu$ is too large, another
difficulty arises for the effectively Hamiltonian. For graphene in the regime
of visible light, it was shown [117] that the next nearest neighbor hopping
leads to only small corrections to the Dirac cone, due to the fact that the
nearest hopping energy in graphene is high ($t\approx 2.7$ eV), which
corresponds to a photon of frequency of several hundred Terahertz. For the
pseudospin-1 Lieb lattice, experiments showed that the nearest neighbor
hopping energy is smaller than that in graphene [27], but for the dice lattice
there has been no experimental result.

Figure 2 shows the optical conductivity for different values of $\alpha$ at a
finite temperature. The real part of the conductivity is exemplified in Figs.
2(a), 2(c), and 2(e), to which the intraband process has no contribution.
There are two interband transition points, one is the cone-to-cone transition
that occurs for $\hbar\omega/\mu>2$ and another is the flat-band-to-cone
transition that occurs for $\hbar\omega/\mu>1$. (Note that, for $\alpha=0$,
there is no flat band, so the latter transition does not exist.) For
$\alpha=1$, only the flat-band-to-cone transition exists and its magnitude is
twice of that of the cone-to-cone transition for $\alpha=0$. For the hybrid
lattice, both types of transitions coexist. A finite temperature tends to
smooth the transitions. Figures 2(b), 2(d), and 2(f) show the imaginary part
of the conductivity, where the intraband process gives a singularity at
$\omega\rightarrow 0$, and each interband transition leads to a dip for Im
$(\sigma)<0$, which is smoothed at finite temperatures. A previous study for
graphene demonstrated that, when the imaginary part of the conductivity
becomes negative, a new TE mode can emerge [66]. Physically, it would be
interesting to investigate the significance of a negative imaginary part of
the optical conductivity for the $\alpha$-$\mathcal{T}_{3}$ lattice.

Figure 2: Real and imaginary parts of the optical conductivity of the
$\alpha$-$\mathcal{T}_{3}$ lattice derived from the Kubo formula in the
absence of any impurity scattering. (a,c,e) Real part of the optical
conductivity for $\alpha=0$ (graphene), $\alpha=1/\sqrt{3}$, and $\alpha=1$
(pseudospin-1), respectively. In the zero temperature limit $T\rightarrow 0$,
the conductivity is non-zero for $\hbar\omega/\mu>2$. An interband transition
leads to a dip in the conductivity plot. The step-function type of transition
is smoothed out by finite temperatures. (b,d,f) Imaginary part of the optical
conductivity for $\alpha=0$, $1/\sqrt{3}$, and 1, respectively.

##  III Intrinsic plasmon modes in $\alpha$-$\mathcal{T}_{3}$ lattice

An important manifestation of the electromagnetic response of 2D Dirac
materials is the intrinsic plasmon modes. Specifically, for 2D materials on a
dielectric substrate, the Maxwell’s equations can be solved analytically,
where the propagating modes follow certain dispersion relationship which
depends on the polarization. A solution method was developed earlier [118,
119] and later adopted to graphene [66, 67]. The propagating modes determine
the electromagnetic response of the 2D material and they are also termed
“intrinsic plasmon modes” [59]. Recently, the study of the plasmon modes has
been extended to other kinds of materials such as bilayer graphene [85],
twisted bilayer graphene [88], silicene [93], and hBN [84].

Here we study the intrinsic plasmon modes for $\alpha$-$\mathcal{T}_{3}$
lattice. Suppose we embed the lattice in-between two materials with dielectric
constants $\epsilon_{1}$ and $\epsilon_{2}$, respectively, as illustrated in
Fig. 1(c). The wavevector $q$ is in the $(x,y)$-plane and decays along the $z$
direction. Neglecting the thickness of the lattice and matching the boundary
conditions at $z=0$, we obtain the polarization-dependent (e.g., TM or TE)
dispersion relation [67, 93]. In particular, for the TM polarization, we have

$1+\frac{2\pi i\sigma(\omega,\phi)\sqrt{q^{2}-\omega^{2}/c^{2}}}{\omega}=0,$
(8)

where $\omega$ is the frequency of the incident field and $c$ is the speed of
light. The solution exists for Im $(\sigma)>0$. For 2D materials, the
conductivity for small incident frequency is described by the Drude model
[118, 119]. For TE polarization, the dispersion relation is

$1-\frac{2\pi
i\omega\sigma(\omega,\phi)}{c^{2}\sqrt{q^{2}-\omega^{2}/c^{2}}}=0,$ (9)

where the solution exists for Im $(\sigma)<0$. As revealed by previous work on
graphene [66], novel TE polarization can arise for $1.667<\hbar\omega/\mu<2$.
This can be seen from Fig. 2(b), where the interband transition leads to non-
zero negative imaginary conductivity.

TE polarization is a unique feature originated from the linear dispersion
relationship, which arises for relatively high frequencies. Previous work
revealed that the TE transition can be exploited to develop graphene-based
polarizer in the visible light regime [71] and high-frequency optical switches
[72]. The negative imaginary conductivity leading to TE polarization was
exploited to construct a superscattering system with a curved copper-coated
cylinder [78], where a large scattering cross section was obtained even when
the dimension of the scatterer is much smaller than the wavelength.

The solution of Eqs. (8) and (9) exist for nonzero imaginary conductivity, but
not all the solutions correspond to propagating modes. To characterize wave
propagation in the lattice, we use the loss and attenuation length. In
particular, the loss is defined as

$\text{Loss}\equiv\frac{\text{Im}(q)}{\text{Re}(q)},$ (10)

which determines the average propagation length in the 2D lattice (a small
loss leads to longer propagation). The attenuation length or skin depth is
defined as [67]

$\xi=\frac{k}{2\pi
k_{z}}=\frac{\lambda_{z}}{2\pi\lambda}=\frac{\omega/c}{2\pi\text{Re}(\sqrt{q^{2}-(\omega/c)^{2}})},$
(11)

where $k_{z}$ is a wave vector measuring the confinement in the $z$ direction
perpendicular to the lattice plane. The strength of the wave in the $z$
direction is proportional to $\exp(-k_{z}|z|)$ and

$\lambda_{z}=2\pi/k_{z}=\text{Re}(2\pi/\sqrt{q^{2}-(\omega/c)^{2}}).$

A small value of $\lambda_{z}$ corresponds to strong decay in the $z$
direction so the wave is well localized in the lattice plane. It is often
desired to have a small loss and strong confinement at the same time, but
there is a trade-off. In general, TM waves tend to have strong confinement
with a large loss, but TE waves have a small loss with weak confinement [67].

We calculate the loss and confinement properties for electromagnetic wave
propagation in the $\alpha$-$\mathcal{T}_{3}$ lattice and compare with those
of graphene. We consider a finite temperature (e.g., $k_{B}T/\mu=0.01$ or
$k_{B}T/\mu=0.05$) and make the conductivity dimensionless through
$\tilde{\sigma}=\sigma/\sigma_{0}$, where $\tilde{\sigma}$ is decomposed into
a real and an imaginary parts:
$\tilde{\sigma}=\tilde{\sigma}^{\prime}+i\tilde{\sigma}^{\prime\prime}$. For
the TM wave, inserting the conductivity formula into Eq. (8), we get

$q_{\text{TM}}=\frac{\omega}{c}\sqrt{1-\left(\frac{2}{\pi\alpha_{0}\tilde{\sigma}}\right)^{2}},$
(12)

where $\alpha_{0}\approx 1/137$ is the fine structure constant. The small
denominator in the second term leads to a large imaginary value of
$q_{\text{TM}}$, so the loss is large for TM waves. For TE waves, we have

$\begin{split}q_{\text{TE}}=&\frac{\omega}{c}\sqrt{1-\left(\frac{\pi\alpha_{0}}{2}\tilde{\sigma}\right)^{2}}\\\
\approx&\frac{\omega}{c}\left[1-i\left(\frac{\pi\alpha_{0}}{2}\right)^{2}\tilde{\sigma}^{\prime}(-\tilde{\sigma}^{\prime\prime})\right],\end{split}$
(13)

where $\tilde{\sigma}$ is dimensionless and $\alpha_{0}$ is small so the
Taylor expansion has been used for the square root. Since the loss is
proportional to $\alpha_{0}^{2}$, it is small for TE waves.

Figure 3: Loss and attenuation length for TM wave propagation in the
$\alpha$-$\mathcal{T}_{3}$ lattice. (a,c,e) Loss versus normalized frequency
for $\alpha=0$, $1/\sqrt{3}$ and 1, respectively. (b,d,f) The respective
attenuation length versus normalized frequency. In each panel, the two curves
correspond to two different temperatures: $T/\mu=0.01$ or $T/\mu=0.05$,
respectively.

The attenuation lengths for the TM and TE waves are

$\xi_{\text{TM}}=\frac{\alpha_{0}}{4}\frac{|\tilde{\sigma}|^{2}}{\tilde{\sigma}^{\prime\prime}}\
\ \text{and}\ \
\xi_{\text{TE}}=\frac{1}{\pi^{2}\alpha_{0}\tilde{\sigma}^{\prime\prime}}.$
(14)

We first study TM wave propagation in $\alpha$-$\mathcal{T}_{3}$ lattice for
different $\alpha$ values. Since TM polarization occurs only for
$\mbox{Im}(\sigma)>0$, the underlying waves arise for small $\omega$ values
for which the intraband transition dominates. Figures 3(a), 3(c) and 3(e) show
the loss versus the incident frequency for three different $\alpha$ values,
respectively, from Eqs. (10) and (12). The loss is small when there is strong
intraband transition ($\omega\rightarrow 0$). The loss becomes large when
interband transition is about to happen. As the temperature increases, the
loss become larger but this effect is insignificant. Figures 3(b), 3(d) and
3(f) display the attenuation length for TM waves. From Eq. (14), we have that,
if the conductivity is purely imaginary, the attenuation length is
proportional to $\tilde{\sigma}^{\prime\prime}$. As a result, for
$\omega\rightarrow 0$ the attenuation length is large. As the temperature
increases, the sharp dips in the attenuation length are smoothed out. As the
value of $\alpha$ increases, the flat-band-to-cone transition begins to
dominate, making the imaginary part of the conductivity negative for small
$\omega$, so the viable region for TM wave propagation becomes smaller.

Figure 4: Loss and attenuation length for TE wave propagation in
$\alpha$-$\mathcal{T}_{3}$ lattice. (a,c,e) Loss versus normalized frequency
for $\alpha=0$, $1/\sqrt{3}$ and 1, respectively. (b,d,f) The respective
attenuation length versus the normalized frequency. In each panel, the two
curves correspond to two different temperatures: $T/\mu=0.01$ or $T/\mu=0.05$,
respectively. The shaded regions indicate the possible windows for TE wave
propagation. For $\alpha=0$ (a,b), the region is $\hbar\omega/\mu\in(1.67,2)$.
For $\alpha=1/\sqrt{3}$ (c,d), the region is
$\hbar\omega/\mu\in(0.9,1.1)\cap(1.67,2)$. For $\alpha=1$ (e,f), the region is
$\hbar\omega/\mu\in(0.85,1)$.

Next we study TE wave propagation in the $\alpha$-$\mathcal{T}_{3}$ lattice
according to Eqs. (10) and (13). Figures 4(a), 4(c) and 4(e) show the loss
versus the incident frequency for three different values of $\alpha$,
respectively, and the corresponding results for the attenuation length are
shown in Figs. 4(b), 4(d) and 4(f). The loss is smaller than $10^{-4}$ due to
the small imaginary part. After interband transition arises, the loss
increases due to the finite positive real value of the conductivity. From Eq.
(14), we see that the attenuation length is inversely proportional to
$\tilde{\sigma}^{\prime\prime}$, so at the transition point the wave is
maximally localized. Note that, as $\hbar\omega\rightarrow\infty$, even when
the loss decreases, the attenuation length increases, inhibiting the
propagating wave. In each panel of Fig. 4, the shaded regions represent those
with relatively low loss and small attenuation length. For example, for
$\alpha=0$, this occurs before the cone-to-cone transition, i.e., for
$\hbar\omega/\mu\in(1.67,2)$, which agrees with the previous result for
graphene [66]. For $\alpha=1/\sqrt{3}$, there are two shaded regions due to
the coexistence of two transitions. For $\alpha=1$, there is one shaded region
due to the flat-band-to-cone transition. A higher temperature can lead to an
increase in both the loss and attenuation length.

To summarize these results briefly, we have that, for TM waves, the loss is
high - typically about $0.1$ to $1$ but the attenuation length is small, so
the TM waves are highly localized with high loss. Because of the positive
imaginary part of the conductivity $\sigma(\omega,\phi)$, as $\alpha$
increases the propagation region shrinks. For TE waves, the loss is low -
typically less than $10^{-4}$ but the attenuation length is large, so the
waves are weakly localized. As $\alpha$ increases from zero, the bandwidth for
propagation first increases due to the coexistence of multi-interband
processes. For $\alpha=1$, the flat-band-to-cone transition has a large
magnitude and is the only process present.

There are two unique features of electromagnetic wave propagation in the
$\alpha$-$\mathcal{T}_{3}$ lattice. First, comparing panels (a,b) with (e,f)
in Fig. 4, we have that the TE waves for pseudospin-1 lattice have a narrow
frequency width and a small attenuation length, indicating that the intrinsic
plasmon mode is strongly localized, which is due to the transition from the
flat-band to the linear band. Second, Figs. 4(c) and 4(d) indicate that
multiple frequency TE polarization waves can arise: one is the same as
graphene at $\hbar\omega/\mu\approx 2$ due to the cone-to-cone transition and
another occurs at $\hbar\omega/\mu\approx 1$, which is due to the flat-band-
to-cone transition. For $\mu=0.5$ eV, the linear dispersion relation holds, so
the regions that can support TE wave propagation correspond to 110–125 THz and
190–240 THz at the room temperature ($k_{B}T/\mu=0.05$ \- about $300$K).

Figure 5:  Emergence of TE polarization waves in $\alpha$-$\mathcal{T}_{3}$
materials. (a,b) Real and imaginary parts of the optical conductivity,
respectively, for $\alpha=0$ (graphene), $\alpha=1/\sqrt{3}$, and $\alpha=1$
(pseudospin-1). The parameters are $\mu=0.5$eV, $T=300$K and $\tau=6.4\times
10^{-13}$. In the low frequency regime $\omega\rightarrow 0$, intraband
transitions dominate so the conductivity for different values of $\alpha$
converges. (c,d) Imaginary part of the optical conductivity for the hybrid
case ($\alpha\neq 0$) at two temperatures ($T=60$K and $T=300$K, respectively)
for $\mu=0.5$eV. The two local minima are associated with two different
interband transitions, which broaden as the temperature increases.

We have studied the effect of impurity scattering on the optical conductivity.
Figures 5(a) and 5(b) display the conductivity curve in the frequency range
$[0,300]$ THz in the presence of finite impurity scattering. The value of the
relaxation time is taken to be that of graphene ($\alpha=0$) [115], as the
experimental value of this time for $\alpha\neq 0$ has not been available yet.
It can be seen that finite impurity scattering can change the real part of the
intraband conductivity and smooth out the $\delta$-function. However, the
change mainly occurs for frequencies below 10THz and is generally
insignificant. In fact, for frequencies above 100THz, finite impurity
scattering has little effect on the optical conductivity. Our formula of the
intraband conductivity also suggests that it depends on the relaxation time
but not on the value of $\alpha$. We note that a similar observation was made
earlier [97].

The interval in $\alpha$ in which the two TE polarization waves arise can be
seen from Figs. 5(c) and 5(d). At low temperatures, there are two distinct
local minima generated by interband transitions for $\alpha\in[0.4,0.6]$. When
the temperature increases to $k_{B}T/\mu=0.05$, the interval shrinks. This
phenomenon can be understood, as follows. According to Eqs. (5) and (6), the
strength of the cone-to-cone transition is proportional to $\cos^{2}(2\phi)$
and that of the flat-band-to-cone transition is proportional to
$2\sin^{2}(2\phi)$, where $\alpha\equiv\tan(\phi)$. Without taking into
consideration intraband transitions, the two types of interband transition
would have the same magnitude for $2\sin^{2}(2\phi)=\cos^{2}(2\phi)$, which
correspond to $\alpha\approx 0.3$. From Eq. (4), intraband transitions give a
positive imaginary value in the conductivity and the interband transitions
generate a negative imaginary value. Since the strength of intraband
transitions is inversely proportional to $\omega$, the effective strength of
the flat-band-to-cone transition is reduced. To have approximately the same
strength for the cone-to-cone and flat-band-to-cone transitions, it is
necessary to increase the value of $\alpha$ to facilitate the flat-band-to-
cone transition. Numerically, the optimal $\alpha$ interval in which the two
types of interband transition are approximately equal can be obtained by
monitoring the respective local minima in the conductivity generated by the
transitions and examining when the two local minima reach a similar magnitude.
From Response Figs. 5(c) and 5(d), we note that the two local minima are
similar for $\alpha\in(0.4,0.6)$. At finite temperatures, the local minima
will be smoothed out, so the interval shrinks as the temperature increases.

##  IV Resonant scattering from an $\alpha$-$\mathcal{T}_{3}$ lattice coated
dielectric sphere

In electromagnetics, for applications such as optical sensing, imaging,
tagging and spectroscopy [120, 121, 122], enhanced scattering is desired.
There are also applications where reduced scattering is sought, such as
cloaking [123, 124], which can be realized through the technique of scattering
cancellation. In particular, by coating an additional material layer outside
the original scatterer [125], destructive interference can be induced between
the scattering waves from the two structures. Not only is the method of
coating capable of inducing cloaking, but it can also enhance scattering [126]
with proper coating materials and design. For example, a dielectric structure
coated with graphene can lead to cloaking at some wavelength [76, 127, 128,
129], but at other wavelength optical scattering can be enhanced [77]. These
behaviors can be controlled by tuning the chemical potential.

For scattering from a graphene coated structure, the surface conductivity will
generate certain polarization. The change in the scattering cross section is
strongly related to the intrinsic plasmon modes. For plasmon modes with TM
polarization, superscattering or cloaking typically arise in the 1–10 THz
range due to intraband transitions [76, 127, 128]. For TE waves, the frequency
is higher and is strongly modulated by the temperature [77]. Due to the small
imaginary part of the conductivity, scattering is typically weak. Multilayer
structures can be used to enhance the polarization [71, 84, 95]. A previous
experiment demonstrated that, for graphene, scattering structures with up to
five layers can be designed, with scattering loss comparable to that with the
monolayer structure [71].

Here we study optical scattering from a dielectric sphere coated by multiple
layers of $\alpha$-$\mathcal{T}_{3}$ lattice. For a multilayer lattice, the
dielectric constant (relative permittivity) is defined as [71, 128]

$\epsilon_{\alpha,N}=1+i\frac{N\sigma(\omega,\phi)}{r_{2}\epsilon_{0}\omega},$
(15)

where $N$ is the number of layers, $\epsilon_{0}$ is vacuum permittivity, and
$r_{2}$ is the total depth of the coating structure whose value is usually
chosen to be much smaller than the device dimension to reduce the finite-depth
effect [128]. We choose a dielectric sphere of radius $r_{1}=100$ nm with
dielectric constant $\epsilon_{1}=2.1$, corresponding to materials such as
Polytetrafluoroethylene (PTFE) [130], and set $r_{2}=1\ \text{nm}\ll r_{1}$.
Computationally, the multilayer lattice structure can be treated as a single
layer, as all that is required for calculating the electromagnetic scattering
cross section is the conductivity of the $\alpha$-$\mathcal{T}_{3}$ lattice.
Once the dimension and frequency dependent dielectric constant
$\epsilon_{\alpha,N}$ is given, the scattering system can be analytically
solved with proper solutions of the Maxwell’s equations.

Figure 6:  Scattering cross section from a dielectric sphere coated with
multiple layers of $\alpha$-$\mathcal{T}_{3}$ lattice. Shown is the ratio
between the scattering cross sections with and without coating ($\sigma$ and
$\sigma_{0}$, respectively) versus frequency. For $\alpha=0$ there is only one
resonance peak at $f\approx 130$ THz. For $\alpha=1/\sqrt{3}$, there are two
peaks at $f\approx 130$ and $f\approx 260$, respectively. For $\alpha=1$ there
is only one peak at $f\approx 130$ and the value of NSCS nearly doubles as
compared with the case of graphene coating. Figure 7: Examples of scattering
wavefunction. Shown are the real (left column) and imaginary (right column)
parts of the scattering field for three cases: (a,b) without coating where the
horizontal line segment has the length $\lambda/10$ and indicates that the
wavelength is much larger than the size of the scatterer, (c,d) graphene
coating and (e,f) pseudospin-1 lattice coating. The scattering field is
relatively strong with the pseudospin-1 lattice coating.

The incident wave is chosen to propagate in the $z$ direction and polarized
along the $x$ direction, and the scattering wave is calculated by using the
transfer matrix method [131, 132]. After decomposing the scattering wave into
a series of spherical Bessel functions, the coefficient for each basis is
obtained. The scattering cross section is given by

$\sigma_{\text{sc}}=\sum_{i}(2i+1)(|c^{\text{TM}}_{i}|^{2}+|c^{\text{TE}}_{i}|^{2})$
(16)

where $c^{\text{TM}}_{i}$ and $c^{\text{TE}}_{i}$ are the coefficients for
scattering wave with the corresponding polarization. (The calculation details
are presented in Appendix C.) To characterize the change in the scattering
cross section before and after coating, we define the normalized scattering
cross section as the ratio between the cross sections with and without the
coated structure:

$\text{NSCS}=\frac{\sigma_{\text{sc coated}}}{\sigma_{\text{sc uncoated}}}.$
(17)

Figure 6 shows NSCS versus frequency for different scattering structures. For
a meaningful comparison, we choose the number of layers to be $N=5$ and set
the temperature to be $T=300$ K for all structures. Without coating, the value
of NSCS is one. For $\alpha=0$ (graphene), when the chemical potential of
$0.25$ eV is applied, there is a resonance at $f\approx 130$ THz, which is
approximately $\hbar\omega\approx 2\mu$ and corresponds to a TE intrinsic
plasmon mode in graphene. For $\alpha=1/\sqrt{3}$, two resonance peaks arise
but their strength is weak, which correspond to the two different band
transition processes. Similar to the multiple frequency phenomenon associated
with propagating wave, the resonances are temperature-dependent. For
$\alpha=1$, we double the chemical potential for comparing with the case of
graphene, which results in a resonance at the same frequency. The resonance
for $\alpha=1$ can be attributed to the flat-band-to-cone transition, which is
strong due to the large imaginary part of the conductivity.

Figure 7 shows the scattering field for $f=127$ THz, corresponding to the left
resonant peak in Fig. 6, for three cases: without coating, graphene coating,
and pseudospin-1 lattice coating. Without coating, as shown in Figs. 7(a) and
7(b), the scattering field is weak because the wavelength is much larger than
the dimension of the scatterer. Figures 7(c) and 7(d) are for $\alpha=0$ with
the same parameter setting as in Fig. 6. The scattering field for $\alpha=1$
is shown in Figs. 7(e) and 7(f), demonstrating that the field is enhanced as
compared with the case of graphene and greatly enhanced as compared with the
case without coating.

##  V Discussion

Two-dimensional Dirac materials with a flat band have become an active
frontier in condensed matter physics and materials science, and it is of
interest to understand the optical responses of these materials. In this
regard, the basic physical quantity is the optical conductivity. Previous
studies focused on the real part of the conductivity. Utilizing
$\alpha$-$\mathcal{T}_{3}$ lattice as a vehicle, we have derived a complete
formula for the optical conductivity which includes both the real and
imaginary parts. The base of our derivation is the Kubo formula and we have
also exploited the Kramers–Kronig relation to provide an alternative
derivation that leads to the same conductivity formula. Physically, the
presence of a flat band, together with the Dirac cone pairs, gives rise to
richer transitions as compared with Dirac materials without a flat band, such
as graphene. We have demonstrated that the conductivity of
$\alpha$-$\mathcal{T}_{3}$ lattice has three components, each contributed to
by a distinct optical transition. Specifically, the contribution from
intraband transitions is similar to that for Dirac materials without a flat
band. While the cone-to-cone transition is pronounced for $\alpha=0$
(graphene), it weakens as $\alpha$ increases towards one. For $\alpha>0$, the
flat-band-to-cone transition is dominant. For $\alpha=1$ where the influences
of the flat band are the strongest possible, the contribution to the
conductivity from the cone-to-cone transition becomes negligible as compared
with that from the flat-band-to-cone transition.

With the derived frequency-dependent conductivity formula for
$\alpha$-$\mathcal{T}_{3}$ lattice, we have investigated two problems that are
fundamental for developing $\alpha$-$\mathcal{T}_{3}$ based optical devices.
The first is electromagnetic wave propagation, which results in intrinsic
plasmon modes in the $\alpha$-$\mathcal{T}_{3}$ lattice, whose physical
properties depend on the polarization. Our calculations have revealed that TM
polarized waves are the result of intraband transitions, which occur in the
frequency range 1–10 THz. These waves also arise in other 2D Dirac materials
such as graphene. In contrast, TE polarized waves are generated by interband
transitions, which arise in a higher frequency range: 100–300 THz. For
$0<\alpha<1$, two interband transitions occur simultaneously, which generate
two TE surface waves at $\hbar\omega/\mu\approx 1,2$, respectively. This is
verified by calculating the loss and attenuation length that are minimized in
the frequency region. The second problem is scattering from a dielectric
sphere coated with multiple layers of $\alpha$-$\mathcal{T}_{3}$ material. The
finding is that, due to a reduction in the imaginary part of the optical
conductivity at finite temperatures, the scattering of TE polarized waves is
weaker than that of TM waves. The multilayer scattering structure can then be
used to enhance certain polarization [71, 84, 95].

For 2D Dirac materials, an approach to generating scattering or transport
behaviors at multiple frequencies is to use some structure with a special
band, such as bilayer graphene with four bands [85]. For a proper choice of
the chemical potential, the phenomenon of frequency doubling can occur. In
materials similar to graphene such as silicene [93], the sublattice and valley
symmetries are broken, thereby generating multiple energy bands. For these
materials, the electromagnetic property depends on the chemical potential, so
the intrinsic plasmon modes depend on the chemical potential as well. The
frequency tunability of these materials is typically weaker than that of
$\alpha$-$\mathcal{T}_{3}$ materials for $\alpha\lesssim 1$. Recently there is
a growing interest in hexagonal boron nitride (hBN) materials [95] whose
hyperbolic permittivity can lead to a multi-frequency resonant scattering.
However, for the TE polarization in this material, the resonant frequency is
fixed. In contrast, our study here has demonstrated that, for the
$\alpha$-$\mathcal{T}_{3}$ lattice, insofar as the linear dispersion holds, TE
polarized waves can be tuned by adjusting the chemical potential,

A general result is that the optical responses of $\alpha$-$\mathcal{T}_{3}$
materials for $\alpha\lesssim 1$ are in general more pronounced than those for
$\alpha\gtrsim 0$, e.g., graphene, as the conductivity due to the flat-band-
to-cone transition is twice of that due to the cone-to-cone transition. For
the same frequency, this means that for pseudospin-1 materials, the chemical
potential is effectively doubled, making the optical responses twice as strong
as those for graphene. A physical reason for this enhancement is that the
plane wave in the pseudospin-1 lattice has a smaller attenuation length due to
the large imaginary part of the optical conductivity as compared to that in
graphene. From the point of view of resonant scattering, at the same
frequency, a larger scattering cross section can arise in pseudospin-1
materials in comparison with graphene. We note that hBN materials can also
generate stronger surface plasmon waves than graphene [83, 84].

When performing the scattering calculation for a multilayer
$\alpha$-$\mathcal{T}_{3}$ structure, we ignored the effect of interlayer
coupling. This approximation can be justified, as follows. In a study of the
optical conductivity in monolayer and double-layer $\alpha$-$\mathcal{T}_{3}$
lattice [133], it was observed that the coupling decays exponentially with the
interlayer spacing. The results on the plasmon density indicate that only for
small layer spacing will the original peak split into two distinct peaks. For
reasonably large spacing, e.g., $d=5k_{F}^{-1}$, the layer coupling effect can
be neglected.

An extensively studied case is graphene, where monolayer graphene has two
linear bands. The bilayer graphene system studied in Refs. [65, 85] has four
bands due to interlayer coupling and, as a result, there is splitting in the
plasmon peaks. However, when the interlayer distance is large, the coupling
effect becomes negligible. For $v_{F}=10^{6}$ and $\mu=1$ eV, this critical
spacing is about $d=3.3$ nm. In experiments, stacking geometry is often used
to generate multilayer graphene system, where the interlayer coupling effect
is insignificant and the electronic structure of the system is similar to that
of monolayer graphene [71].

We remark on the possible many-body effects. In graphene, the long-range
Coulomb interaction can lead to a renormalized Fermi velocity for $\mu=0$ and
$T=0$. In Ref. [134], a Hartree-Fock theory was developed with no fitting
parameters but with a topological invariant. A theoretical calculation of the
Fermi velocity renormalization agreed with the experimental data. While the
renormalization can change the optical conductivity, the effect is small.

In Ref. [135], a similar renormalization effect was observed in lattices with
a flat band. For $\mu=0$ and a partially filled flat band, the Drude weight
has several zeros depending on the inverse ratio of the band filling. Since
the intraband conductivity is proportional to the Drude weight, such a change
may also lead to conductivity oscillations for $\mu=0$ and small $\omega$. Our
work focuses on the case of a positive chemical potential $\mu$, where the
renormalization effect is small.

Can the flat band contribute to the intraband conductivity? In a previous work
[136], the authors considered a lattice model with a flat band in the presence
of correlated disorders that provide coupling between the flat-band and
dispersion band states. For a finite lattice and an initial Gaussian wave
packet in the real space, some states in the flat band are unoccupied, giving
rise to intraband transitions within the flat band. In our study, correlated
disorders were assumed to be absent and the Fermi energy is positive so that
the flat band is fully occupied for $T\rightarrow 0$. Since only the states
near the Fermi surface are excited, the flat band gives no contribution to the
intraband conductivity. The same observation was made in another previous
study [99], where the real part of the conductivity was derived with the
finding that the intraband conductivity does not depend on $\alpha$.

With regard to impurity scattering, previously the imaginary part of
conductivity was studied in the low frequency regime [111, 112, 113]. It was
found that the relaxation time $\tau$ has a significant effect on the
convergence of the conductivity. For a small impurity scattering rate, the
imaginary part tends to diverge for near zero frequency. The values of the
relaxation time for flat-band Dirac materials are not known at the present.
However, our study focuses on the physically more relevant high frequency
regime.

To summarize, the analysis of our complete formula of the optical conductivity
for $\alpha$-$\mathcal{T}_{3}$ suggest a number of phenomena that can be
useful for designing optical devices based on this type of generalized 2D
Dirac materials. For example, for $\alpha\in(0.4,0.6)$, because of the
coexistence of cone-to-cone and flat-band-to-cone transitions, the
corresponding lattice can be exploited for TE wave based broad-band devices.
Another phenomenon is that the magnitude of the optical conductivity of
pseudospin-1 materials is twice as large as that of graphene due to a
reduction in the energy required for a flat-band-to-cone transition. A strong
TE wave can then be generated and sustain at $\hbar\omega/\mu\approx 1$,
giving the possibility to design optical devices using
$\alpha$-$\mathcal{T}_{3}$ ribbon or other coupling structures [74, 127, 137].
Recent work in quantum plasmonics has suggested that edge states in graphene
can lead to a blue shift in the plasmon modes [138]. It would be interesting
to exploit $\alpha$-$\mathcal{T}_{3}$ materials for applications in quantum
plasmonics.

## Acknowledgement

This work was supported by AFOSR under Grant No. FA9550-21-1-0186.

##  Appendix A Optical matrix for $\alpha$-$\mathcal{T}_{3}$ lattice

In the tight-binding framework, the low energy excitations in the
$\alpha$-$\mathcal{T}_{3}$ lattice are described by the Hamiltonian (1).
Associated with the first valley ($v=1$), there are three bands: $0,\pm 1$,
corresponding to the flat band, conduction and valence bands, respectively.
The eigenfunctions are

$|\psi_{\pm 1}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos\phi
e^{i\theta_{\mathbf{k}}}\\\ \pm 1\\\ \sin\phi
e^{-\theta_{\mathbf{k}}}\end{pmatrix}$ (18)

and

$|\psi_{0}\rangle=\begin{pmatrix}\sin\phi e^{i\theta_{\mathbf{k}}}\\\ 0\\\
-\cos\phi e^{-i\theta_{\mathbf{k}}}\end{pmatrix},$ (19)

where $f_{\mathbf{k}}=|f_{\mathbf{k}}|e^{i\theta_{\mathbf{k}}}$ and
$\theta_{\mathbf{k}}$ is the angle of $\mathbf{k}$ in the polar coordinates.
For the second valley, we have $f_{\mathbf{k},v=-1}=-f_{\mathbf{k},v=1}^{*}$,
so the solutions can be obtained from a sign change:
$\theta_{\mathbf{k}}=-\theta_{\mathbf{k}}$. Because of the mirror symmetry in
the integration with respect to $\mathbf{k}$, the minus sign will not change
the results.

The optical matrix elements associated with the current operator in the $x$
direction are given by

$\begin{split}|\langle\mathbf{k},\pm|j_{x}|\mathbf{k},\pm\rangle|^{2}&=e^{2}v_{F}^{2}\cos^{2}\theta_{\mathbf{k}}\\\
|\langle\mathbf{k},\pm|j_{x}|\mathbf{k},\mp\rangle|^{2}&=e^{2}v_{F}^{2}\sin^{2}\theta_{\mathbf{k}}\cos^{2}(2\phi)\\\
|\langle\mathbf{k},0|j_{x}|\mathbf{k},\pm\rangle|^{2}&=|\langle\mathbf{k},\pm|j_{x}|\mathbf{k},0\rangle|^{2}\\\
&=\frac{e^{2}v_{F}^{2}}{2}\sin^{2}\theta_{\mathbf{k}}\sin^{2}(2\phi).\end{split}$
(20)

##  Appendix B Optical conductivity of $\alpha$-$\mathcal{T}_{3}$ lattice

###  B.1 Derivation based on the Kubo formula

With the Kubo formula (3), the summation over different states can be
simplified to the summation from $k=k^{\prime}$, which corresponds to the
direct optical transition. There are three types of band transitions.

#### Intraband transition.

The transition is from the conduction band to itself with
$E_{n}-E_{m}\rightarrow 0$ and $E_{n}\approx E_{m}\approx\mu$. Under the
approximations, the Fermi-Dirac distribution can be written as

$\frac{F(E_{m})-F(E_{n})}{E_{n}-E_{m}}=-\left.\frac{\partial
F}{\partial\epsilon}\right|_{\epsilon=\mu}=\delta(\epsilon-\mu).$ (21)

Equation (3) then becomes

$\sigma^{(1)}(\omega,\phi)=\frac{\hbar}{i\pi^{2}}\iint
dk_{x}dk_{y}\frac{\partial
F}{\partial\epsilon}\frac{j_{nm}^{2}}{\hbar\omega}.$ (22)

Inserting the optical matrix elements into Eq. (20) and using the linear
dispersion relationship $E=\hbar v_{F}|\mathbf{k}|$, in the polar coordinates
we have

$\iint dk_{x}dk_{y}j_{nm}^{2}=\frac{e^{2}}{\hbar^{2}}\int_{0}^{\infty}\epsilon
d\epsilon\int_{0}^{2\pi}\cos^{2}\theta_{\mathbf{k}}d\theta_{\mathbf{k}}.$ (23)

Equation (22) becomes

$\sigma^{(1)}(\omega,\phi)=\frac{e^{2}}{i\pi\hbar^{2}\omega}\int\epsilon[-\delta(\epsilon-\mu)]d\epsilon=\frac{ie^{2}\mu}{\pi\hbar^{2}\omega}.$
(24)

Introducing $\sigma_{0}=e^{2}/(4\hbar)$, we obtain the intraband conductivity
as

$\sigma^{(1)}(\omega,\phi)=\frac{4i\mu\sigma_{0}}{\pi\hbar\omega}.$ (25)

#### Cone-to-cone transition.

This transition occurs from $|-\rangle$ to $|+\rangle$ and vice visa. Due to
the involvement of two different bands, an additional factor of $2$ arises in
the summation;

$\begin{split}\sigma^{(2)}(\omega,\phi)=&\frac{\hbar}{i\pi^{2}}\sum_{n,m}\frac{F(E_{m})-F(E_{n})}{E_{n}-E_{m}}\times\\\
&\frac{j_{nm}^{2}(-2\hbar\omega)}{(\hbar\omega)-(E_{n}-E_{m})^{2}}.\end{split}$
(26)

Since $\mathbf{k}=\mathbf{k}^{\prime}$ and because $E_{n}$ and $E_{m}$ belong
to different bands, we can write $E_{n}=\epsilon$ and $E_{m}=-\epsilon$. Using
the integral in Eq. (23) and the optical matrix elements Eq. (20), we get

$\begin{split}\sigma^{(2)}(\omega,\phi)=&\cos^{2}(2\phi)\frac{e^{2}}{i\pi\hbar}\int[F(-\epsilon)-F(\epsilon)]\times\\\
&\frac{\hbar\omega}{4\epsilon^{2}-(\hbar\omega)}d\epsilon.\end{split}$ (27)

The difference in the Fermi-Dirac distribution from the case of intraband
transition implies that a non-zero value occurs only for $\epsilon>\mu$ or
$\epsilon<-\mu$, so in the polar coordinates only the first term is
meaningful. We obtain

$\sigma^{(2)}(\omega,\phi)=\cos^{2}(2\phi)\frac{e^{2}}{i\pi\hbar}\int_{\mu}^{\infty}\frac{\hbar\omega}{4\epsilon^{2}-(\hbar\omega)^{2}}d\epsilon.$
(28)

This integral has a singularity for $2\hbar\omega>\mu$. Using the residue
theorem, we get

$\begin{split}\sigma^{(2)}(\omega,\phi)=&\cos^{2}(2\phi)\sigma_{0}\times\\\
&\left[\Theta(\hbar\omega-2\mu)-\frac{i}{\pi}\ln\left|\frac{\hbar\omega+2\mu}{\hbar\omega-2\mu}\right|\right],\end{split}$
(29)

where $\Theta$ is the Heaviside step function. It can be verified that, for
$\phi=0$, Eq. (29) reduces to the formula for graphene. For $\phi=\pi/4$, the
integral is zero, indicating that for the pseudospin-1 lattice, the cone-to-
cone transition has no contribution to the optical conductivity.

#### Flat-band-to-cone transition.

The derivation of the contribution to the optical conductivity by the flat-
band-to-cone transition is similar to that with the cone-to-cone transition.
In particular, for the flat-band-to-cone transition, we have $E_{n}=0$ and
$E_{m}=\epsilon$, so

$\sigma^{(3)}(\omega,\phi)=\sin^{2}(2\phi)\frac{e^{2}}{i\pi\hbar}\int_{\mu}^{\infty}\frac{\hbar\omega}{\epsilon^{2}-(\hbar\omega)^{2}}d\epsilon.$
(30)

The singularity now occurs at $\hbar\omega=\epsilon$ with the weight
$\sin^{2}(2\phi)$. Evaluating this integral, we get

$\begin{split}\sigma^{(3)}(\omega,\phi)=&2\sin^{2}(2\phi)\sigma_{0}\times\\\
&\left[\Theta(\hbar\omega-\mu)-\frac{i}{\pi}\ln\left|\frac{\hbar\omega+\mu}{\hbar\omega-\mu}\right|\right].\end{split}$
(31)

###  B.2 Derivation based on Kramers-Kronig formula

As proposed in Ref. [88], a different approach to deriving the optical
conductivity is to calculate the real part first and then use the Kramers-
Kronig formula to obtain the imaginary part. We express the conductivity as
the sum of a “normal” term and a term containing a $\delta$-singularity

$\sigma(\omega,\phi)_{\text{tot}}=\sigma(\omega,\phi)+\pi
D\delta(\hbar\omega),$ (32)

where $D$ is the Drude weight (or charge stiffness, to be defined below). To
obtain the real part of the “normal” term, we use

$\begin{split}&\text{Re}[\sigma(\omega,\phi)]=\frac{1}{2\pi\omega}\iint
dk_{x}dk_{y}\sum_{n,m}\\\
&[F(E_{m})-F(E_{n})]j_{nm}^{2}\delta[\hbar\omega-(E_{n}-E_{m})].\end{split}$
(33)

First, we consider the cone-to-cone transition where $E_{n}=\epsilon$,
$E_{m}=-\epsilon$, and the band degeneracy is two. We have

$\begin{split}\text{Re}[\sigma^{(2)}&(\omega,\phi)]=\cos^{2}(2\phi)\frac{e^{2}}{\pi\omega\hbar^{2}}\int_{0}^{2\pi}\cos^{2}\theta_{\mathbf{k}}d\theta_{\mathbf{k}}\times\\\
&\int_{0}^{\infty}\epsilon[F(-\epsilon)-F(\epsilon)]\delta(\hbar\omega-2\epsilon)d\epsilon.\end{split}$
(34)

The difference in the Fermi-Dirac functions is one only for $\epsilon>\mu$,
and the $\delta$ function is contained in the integration region for
$2\hbar\omega>\mu$. We thus have a step transition at $\mu=2\hbar\omega$:

$\text{Re}[\sigma^{(2)}(\omega,\phi)]=\cos^{2}(2\phi)\sigma_{0}\Theta(\hbar\omega-2\mu).$
(35)

Similarly, we obtain, for the flat-band-to-cone transition, the real part of
the conductivity:

$\begin{split}\text{Re}[\sigma^{(3)}(\omega,\phi)]&=\sin^{2}(2\phi)\frac{e^{2}}{2\pi\omega\hbar^{2}}\int_{0}^{2\pi}\sin^{2}\theta_{\mathbf{k}}\times\\\
&\int_{0}^{\infty}\epsilon[f(0)-f(\epsilon)]\delta(\hbar\omega-\epsilon)d\epsilon\\\
&=2\sin^{2}(2\phi)\sigma_{0}\Theta(\hbar\omega-\mu),\end{split}$ (36)

which is consistent with the result in Ref. [97].

To calculate the imaginary part of the conductivity, we use the Kramers–Kronig
(KK) formula [139], which connects the real and imaginary parts of the
response function. There is an additional term in the integral, which is
determined by a cutoff in the case of twisted bilayer graphene [88]. The
method consists of three steps.

_Step 1_ : Calculate the maximum of Re $(\sigma)$ for
$\omega\rightarrow\infty$:

$\sigma_{m}(\phi)=\lim_{\omega\rightarrow\infty}\text{Re}[\sigma(\omega,\phi)].$
(37)

_Step 2_ : Define the Drude weight (or charge stiffness) as

$D=\lim_{\Lambda\rightarrow\infty}\frac{2}{\pi}\left(\sigma_{m}(\phi)\Lambda-\int_{0}^{\Lambda}\text{Re}[\sigma(\nu,\phi)]d\nu\right).$
(38)

_Step 3_ : Use the Kramers-Kronig relation to write the imaginary part of the
conductivity as

$\begin{split}\text{Im}[\sigma(\omega,\phi)]=&\frac{D}{\hbar\omega}+\\\
&\frac{2\hbar\omega}{\pi}\mathcal{P}\int_{0}^{\infty}\frac{\text{Re}[\sigma(\nu,\phi)]-\sigma_{m}(\phi)}{(\hbar\omega)^{2}-\nu^{2}}d\nu.\end{split}$
(39)

For the $\alpha$-$\mathcal{T}_{3}$ lattice, we have

$\sigma_{m}(\phi)=\sigma_{0}[2\sin^{2}(2\phi)+\cos^{2}(2\phi)],$

so the Drude weight is

$D=\frac{4\sigma_{0}\mu}{\pi},$ (40)

and the Kramers-Kronig relation becomes

$\begin{split}\text{Im}[\sigma(\omega,\phi)]=&\frac{4\sigma_{0}\mu}{\pi\hbar\omega}+\frac{2\hbar\omega}{\pi}\mathcal{P}\int_{0}^{\infty}\frac{\text{Re}[\sigma(\nu,\phi)]-\sigma_{m}(\phi)}{(\hbar\omega)^{2}-\nu^{2}}d\nu\\\
=&\frac{4\sigma_{0}\mu}{\pi\hbar\omega}-\cos^{2}(2\phi)\frac{\sigma_{0}}{\pi}\ln\left|\frac{\hbar\omega+2\mu}{\hbar\omega-2\mu}\right|-\\\
&\sin^{2}(2\phi)\frac{2\sigma_{0}}{\pi}\ln\left|\frac{\hbar\omega+\mu}{\hbar\omega-\mu}\right|.\end{split}$
(41)

Together with the real part in Eqs. (35) and (36) as well as the imaginary
part Eq. (41), we obtain the same conductivity formulas as these derived based
on the Kubo formula [Eqs. (25), (29), and (31)].

###  B.3 Effects of finite temperatures on optical conductivity in the
$\alpha$-$\mathcal{T}_{3}$ lattice

Consider graphene at a finite temperature, where the Fermi-Dirac distribution
can no longer be treated as a step function. Previous work [140] gives

$\begin{split}\Theta(\hbar\omega-2\mu)\rightarrow&\frac{1}{2}+\frac{1}{\pi}\arctan\left[\frac{\hbar\omega-2\mu}{2k_{B}T}\right],\\\
|\hbar\omega-2\mu|\rightarrow&\sqrt{(\hbar\omega-2\mu)^{2}+2(k_{B}T)^{2}}.\\\
\end{split}$ (42)

Comparing Eqs. (29) and (31) as well as the Fermi-Dirac distribution, we can
eliminate the factor of $2$ in Eq. (42) and study the effects of finite
temperature on the flat-band-to-cone transition through the transformations:

$\begin{split}\Theta(\hbar\omega-\mu)\rightarrow&\frac{1}{2}+\frac{1}{\pi}\arctan\left[\frac{\hbar\omega-\mu}{k_{B}T}\right],\\\
|\hbar\omega-\mu|\rightarrow&\sqrt{(\hbar\omega-\mu)^{2}+(k_{B}T)^{2}}.\\\
\end{split}$ (43)

Substituting Eqs. (42-43) into Eqs. (4-6) gives the complete optical
conductivity of $\alpha$-$\mathcal{T}_{3}$ lattice at a finite temperature.

###  B.4 Effects of impurity scattering on optical conductivity in the
$\alpha$-$\mathcal{T}_{3}$ lattice

When impurity scattering occurs, a Drude peak will arise in the intra-band
component of the conductivity. For graphene, a previous work [64] revealed
that the Drude peak gives a singularity at $\omega\rightarrow 0$ of the form
$\mu\delta(\hbar\omega)$. Here we derive the corresponding formula for
$\alpha$-$\mathcal{T}_{3}$ lattice.

Considering the intraband conductivity as given by Eq. (4) and making the
change $\omega\rightarrow\omega+i\tau^{-1}$ with $\tau$ being the relaxation
time, we get

$\sigma^{(1)}(\omega,\phi)=\frac{4i\mu\sigma_{0}}{\pi\hbar(\omega+i\tau^{-1})}.$
(44)

Introducing $\Gamma=\hbar\tau^{-1}$, we have

$\sigma^{(1)}(\omega,\phi)=\frac{4\mu\sigma_{0}}{\pi}\frac{i(\hbar\omega-i\Gamma)}{(\hbar\omega)^{2}+\Gamma^{2}}.$
(45)

Taking two successive limits: first $\hbar\omega\rightarrow 0$ and then
$\Gamma\rightarrow 0$, leads to the Drude peak

$\sigma^{(1)}(\omega,\phi))=4\mu\sigma_{0}\delta(\hbar\omega).$ (46)

Together with Eq. (25), we get Eq. (4) in the main text. This result can be
verified by inserting the Drude weight into Eq. (32).

The above derivation can be justified, as follows. First, the result depends
strongly on the order of limit taking. Note that the Drude singularity is
represented by a $\delta$-function with a coefficient proportional to the
chemical potential $\mu$. However, the function is not properly defined for
$\mu\rightarrow 0$. For graphene in the continuum limit, we have
$\omega\rightarrow 0$ and $\mu\rightarrow 0$, where the conductivity reaches
minimum at the Dirac point but the minimal value remains unresolved [116].
Taking the two limits in the opposite order will generate a different result.
Nevertheless, for optical waves in graphene, because of their high frequency,
a positive Fermi energy is required.

Second, the Drude singularity does not depend on $\alpha$, which is consistent
with the result in Ref. [99]. That is, for different values of $\alpha$, the
intraband conductivity is invariant. Note that, however, the experimental
value of the relaxation time for the $\alpha$-$\mathcal{T}_{3}$ lattice is
currently unavailable.

Third, for finite impurity scattering, the following substitution holds:

$\delta(\hbar\omega)\rightarrow\frac{\Gamma^{2}}{(\hbar\omega)^{2}+\Gamma^{2}}.$
(47)

About the choice of $\tau$, in Ref. [115], the authors suggested
$\tau=6.4\times 10^{-13}$s. The corresponding conductivity is plotted in Figs.
5(a) and 5(b), which reveal that the effect of the impurity on the
conductivity arises only at low frequencies. Heuristically, this is because,
in graphene, $\tau$ is of the magnitude $10^{-13}$s (corresponding
approximately to $10$THz), but optical processes in graphene typically occur
in the frequency regime above this value. As a result, taking into account
impurity scattering will not change our result appreciably.

We note that an alternative approach to obtaining the Drude singularity was
provided in Ref. [88], in which the authors separated the conductivity into
two components: a regular component and a term containing a
$\delta$-singularity:

$\sigma(\omega,\phi)_{\text{tot}}=\sigma(\omega,\phi)+\pi
D\delta(\hbar\omega),$ (48)

where $\sigma$ has been derived in Appendix B2 and $D$ is the Drude weight.
Substituting the formula of $D$ derived in Appendix B2 into the equation leads
to the same result.

###  B.5 Imaginary part of the optical conductivity in the low frequency
regime and the effects of impurity scattering

We study the optical conductivity of the $\alpha$-$\mathcal{T}_{3}$ lattice in
the small frequency regime. To this end, an earlier work considered the
imaginary part of the optical conductivity in superconducting materials in the
zero frequency limit [111, 112], where the results were presented in terms of
the product of the frequency and the imaginary part of the conductivity, which
is related to the inverse square of the penetrate depth [111, 112, 113] as

$\lim_{\omega\rightarrow 0}\omega{\rm Im}\sigma\propto 1/\lambda_{L}^{2}.$

The results indicated that, for a low impurity scattering rate, the quantity
$\omega$Im$\sigma$ can maintain a finite value for a larger frequency interval
near zero. As the impurity scattering rate decreases, the imaginary part of
the conductivity diverges. Physically, this means that the superconducting
materials could have a divergent imaginary part of the conductivity. On the
contrary, for normal states, the quantity $\omega$Im$\sigma$ decays quickly to
zero and thus converges.

Figure 8: Effect of impurity scattering on the optical conductivity of
$\alpha$-$\mathcal{T}_{3}$ lattices in the low frequency regime. Shown is the
quantity $\hbar\omega\rm{Im}\sigma/\sigma_{0}$ (the product between the
imaginary part of the conductivity and the frequency in units of 4eV/$\pi$)
versus the optical frequency $f$ for different values of the material
parameter $\alpha$. In (a), the Fermi energy is 0.5 eV. In this case, the
range of the photon energy (the vertical shaded region in the inset) is far
below the Fermi energy and the flat band states are fully occupied. Interband
transitions are unlikely, so the flat band plays essentially no role in the
optical conductivity. In (b), the Fermi energy is reduced to $0.2$ eV. In this
case, the photon energies as indicated by the shaded region in the inset are
likely to induce interband transitions. The flat band will affect the optical
conductivity, as the conductivity curves are markedly different for different
values of $\alpha$. The effect of impurity scattering is demonstrated in (c),
where the conductivity curves for three values of the scattering relaxation
time $\tau$ are shown for $\alpha=1$. Strong impurity scattering as
characterized by a relatively small value of $\tau$ can lead to a significant
deviation of the conductivity curve from the cases of weaker scattering.

For an $\alpha$-$\mathcal{T}_{3}$ lattice, the two interband transitions as
described by Eqs. (29) and (31) vanish at the low frequency limit, since a low
energy photon is not able to induce an interband transition. For a finite
Fermi energy, impurity scattering does not change this picture. For the
intraband transition in the presence of an impurity, roughly there are two
regimes in Eq. (44). In the first regime, decreasing the frequency $\omega$
will increase the driving period but it is still smaller than the relaxation
time associated with impurity scattering. In this case, it can be seen from
Eq. (44) that the product $\omega$Im$\sigma$ converges to a quantity
proportional to the chemical potential. The second regime is where the
frequency decreases further so that the driving period is longer than the
impurity scattering relaxation time. In this case, impurity scattering
dominates and the imaginary part of the optical conductivity approaches zero.

Figure 8(a) shows $\hbar\omega\rm{Im}\sigma/\sigma_{0}$ versus the frequency
$f$ for $\alpha=0$, $1/\sqrt{3}$ and 1, where the temperature is $T=300$ K and
the chemical potential is $\mu=0.5$ eV. In this frequency range, the
corresponding photon energy lies in the interval as indicated by the shaded
region in the inset. We see that the functional behavior of
$\hbar\omega\rm{Im}\sigma/\sigma_{0}$ versus $f$ is similar for the three
different values of $\alpha$. Since the imaginary part of the optical
conductivity plays an important role in surface wave prorogation and
scattering, the physical significance is that the flat band has little effect
on these processes. In this case, the flat band ($E=0$) states are full, but
any photon energy in the shaded region in Fig. 8(a) is not sufficient to
generate an interband transition. Since the flat band plays no rule in the
optical transitions, different values of $\alpha$ will lead to the same
optical conductivity.

As the Fermi energy is reduced to, e.g., $0.2$ eV, differences in the
functional behavior of $\hbar\omega\rm{Im}\sigma/\sigma_{0}$ versus $f$ begin
to emerge, as shown in Fig. 8(b). From the Fermi-Dirac distribution in the
inset, we see that the photon energy is now sufficient to generate interband
transitions. In this case, the flat band will affect the behavior of
$\hbar\omega\rm{Im}\sigma/\sigma_{0}$, but the effect is reduced for small
frequencies. Increasing the temperature will result in a similar effect.

Figure 8(c) shows $\hbar\omega\rm{Im}\sigma/\sigma_{0}$ versus $f$ for three
values of the relaxation time $\tau$ of impurity scattering. For relatively
large values of $\tau$ (e.g., $\tau=6\times 10^{-12}$ s or $\tau=6\times
10^{-13}$ s), the quantity $\hbar\omega\rm{Im}\sigma/\sigma_{0}$ approximately
attains the value of the chemical potential in the low frequency regime before
decreasing to zero as the frequency increases. As the value of $\tau$
decreases (e.g., to $\tau=6\times 10^{-14}$ s), impurity scattering begins to
dominate, leading to an appreciable change in the functional behavior of
$\hbar\omega\rm{Im}\sigma/\sigma_{0}$ versus $f$. We note that this effect of
impurity scattering on the optical conductivity was previously observed [111].

The results in Fig. 8 can be concisely summarized, as follows.

  * •

For fixed impurity scattering rate and Fermi energy, as $\omega$ decreases,
the product $\omega$Im$\sigma$ first reaches some value proportional to the
chemical potential $\mu$ and then attains a different (but convergent) value
after impurity scattering dominates.

  * •

For fixed impurity scattering rate and frequency $\omega$, the value of
$\omega$Im$\sigma$ depends on $\alpha$ for a small Fermi energy due to the
enhanced interband transition.

  * •

For fixed value of $\alpha$ and Fermi energy, a small impurity scattering rate
(equivalently, a large relaxation time) will lead to divergence of
$\omega$Im$\sigma$. However, for some time $\tau$ that satisfies
$\omega<\tau^{-1}$, the impurity scattering dominates. In this case, the
conductivity is convergent.

##  Appendix C Scattering cross section from a multilayer sphere

To numerically calculate the scattering cross section, we use the iterative
method in Ref. [132]. Here we present the key formulas.

Consider a spherical structure of $L$ layers, where the core is labeled as
$i=1$ and the region outside the structure is denoted as $i=L+1$. In each
layer, the radius is $r_{i}$ and the refractive index is $m_{i}$. Define

$\psi_{n}(\rho)=\rho j_{n}(\rho),\text{ }\xi_{n}(\rho)=\rho
h_{n}^{(1)}(\rho),$ (49)

where $j_{n}$ and $h_{n}^{(1)}$ are the spherical Bessel functions of the
first and third kind, respectively. Consider the following quantities

$\begin{split}D_{n}(\rho)&=\frac{\psi_{n}^{\prime}(\rho)}{\psi_{n}(\rho)},\text{
}G_{n}(\rho)=\frac{\xi_{n}^{\prime}(\rho)}{\xi_{n}(\rho)},R_{n}(\rho)=\frac{\psi_{n}(\rho)}{\xi_{n}(\rho)},\\\
U_{n}(r_{i})&=m_{i}\frac{R_{n}(m_{i}kr_{i})D_{n}(m_{i}kr_{i})+a_{n}^{(i)}G_{n}(m_{k}kr_{i})}{R_{n}(m_{i}kr_{i})+a_{n}^{(i)}},\\\
V_{n}(r_{i})&=\frac{1}{m_{i}}\frac{R_{n}(m_{i}kr_{i})D_{n}(m_{i}kr_{i})+b_{n}^{(i)}G_{n}(m_{k}kr_{i})}{R_{n}(m_{i}kr_{i})+b_{n}^{(i)}},\\\
\end{split}$

where the coefficients $a_{n}^{(i)}$ and $b_{n}^{(i)}$ are to be calculated.
Matching the boundary conditions for each layer leads to the following
iterative equations:

$\begin{split}a_{n}^{(i+1)}&=-R_{n}(m_{i+1}kr_{i})\frac{U_{n}(r_{i})-m_{i+1}D_{n}(m_{i+1}kr_{i})}{U_{n}(r_{i})-m_{i+1}G_{n}(m_{i+1}kr_{i})},\\\
b_{n}^{(i+1)}&=-R_{n}(m_{i+1}kr_{i})\frac{m_{i+1}V_{n}(r_{i})-D_{n}(m_{i+1}kr_{i})}{m_{i+1}V_{n}(r_{i})-G_{n}(m_{i+1}kr_{i})}.\end{split}$

We start from $a_{n}^{(1)}=b_{n}^{(1)}=0$ and iterate the above equations
until the final layer is reached. In our computation, the parameter values are
$m_{1}=\sqrt{\epsilon_{1}}$ and $r_{1}=100$ nm for PTFE,
$m_{2}=\sqrt{\epsilon_{\alpha,N}}$ and $r_{2}=101$ nm for
$\alpha$-$\mathcal{T}_{3}$ lattice, and $m_{3}=1$ for free space. The
scattering cross section is given by

$\sigma_{\text{sc}}=\sum_{n}\frac{2\pi}{k^{2}}(2n+1)\left(|a_{n}^{(3)}|^{2}+|b_{n}^{(3)}|^{2}\right).$
(50)

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