# Plasmon Fizeau drag in 3D Dirac and Weyl semimetals

Morgan G. Blevins  Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Draper
Scholar, Draper, Cambridge, MA 02139, USA  Svetlana V. Boriskina
<EMAIL_ADDRESS>Department of Mechanical Engineering, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA

###### Abstract

There is a need for compact, dynamically tunable nonreciprocal optical
elements to enable on-chip-compatible optical isolators and more efficient
radiative energy transfer systems. Plasmon Fizeau drag, the drag of electrical
current on propagating surface plasmon polaritons, has been proposed to induce
nonreciprocal surface modes to enable one-way energy transport. However,
relativistic electron drift velocities are required to induce appreciable
contrast between the dispersion characteristics of co-propagating and counter-
propagating surface plasmon modes. The high electron drift velocity of
graphene previously allowed for the experimental demonstration of current-
induced nonreciprocity in a two-dimensional (2D) Dirac material. The high
electron drift and Fermi velocities in three-dimensional (3D) Dirac materials
make them ideal candidates for the effect, however, both the theory of the
Fizeau drag effect and its experimental demonstrations in 3D Dirac materials
are missing. Here we develop a comprehensive theory of Fizeau drag in DC-
biased 3D Weyl semimetals (WSM) or Dirac semimetals (DSM), both under local
and non-local approximation and with dissipative losses. We predict that under
practical assumptions for loss, Fizeau drag in the DSM Cd3As2 opens windows of
pseudo-unidirectional transport. We additionally introduce new figures of
merit to rank nonreciprocal plasmonic systems by their potential for
directional SPP transport. Further, we propose a new approach for achieving
appreciable plasmonic Fizeau drag via optically pumping bulk inversion
symmetry breaking WSMs or DSMs.

##  I Introduction

The inherent reciprocity of conventional optical materials imposes fundamental
limits on the efficiency and performance of optical systems such as lasers and
radiative-energy harvesters [1]. Nonreciprocal optical elements prevent bi-
directional wave propagation, allowing propagation of a forward-moving optical
mode while preventing the corresponding backward mode propagation, and thus
enabling the development of optical isolators and circulators [2, 3]. While
magneto-optic materials make up conventional nonreciprocal devices, there is a
growing interest in devices that do not require external magnetic fields,
which can be incompatible with other device materials and require bulky
designs [4]. Alternative approaches via space-time modulation to the material
permittivity [5, 6, 7] and nonlinear optical effects [8, 9, 10, 11] allow for
on-chip integration, but are severely limited by the power consumption
requirements [12].

Another emerging alternative for nonreciprocal optical devices, which requires
neither an external magnetic field nor ultra-fast modulation, is based on
making use of the excitation of surface plasmon polariton (SPP) modes at the
interfaces of Weyl semimetals (WSM) [13, 14, 15, 16, 17]. WSMs are topological
materials with distinct electronic bandstructure, which supports pairs of non-
degenerate, chiral Weyl nodes formed from intersecting, linearly dispersing
bands. WSM must be either inversion symmetry (I) or time-reversal symmetry
(TRS) breaking, or both, to achieve the non-degenerate bandstructure. Due to
the net Berry flux in TRS-breaking WSM, these materials act as nonmagnetic
nonreciprocal materials and have been studied for applications in tunable
near-field radiative heat transfer [18, 16, 15] and as optical isolator
material candidates [19, 20].

Figure 1:  Advantages of Fizeau drag in DC-biased and optically-activated 3D
Dirac and Weyl semimetals for nonreciprocal plasmonics. (a) Plasmon Fizeau
drag via DC bias in the 2D Dirac material, graphene. In 3D material platforms,
DC bias can be applied to (b) noble metals such as Au, Al, etc., or, as we
propose, to (c) a 3D DSM or WSM. (d) The electronic bandstructure of a drift-
biased Weyl/Dirac cone, where the Fermi disk is tilted due to an applied
voltage [21, 22]. (e) Drift-current charge carrier velocities reported for
noble metals and 3D DSMs and WSMs are shown on a log scale in blue. Below, the
ballistic current velocities achievable in WSMs and DSMs via the proposed PGE-
enabled Fizeau drag mechanism are plotted on the same scale in red (data taken
from [23, 24, 25, 26, 27, 28, 29, 30, 31, 32], where DSMs are labeled as
squares and WSMs as diamonds). The proposed optically-activated alternative
mechanism for current drag via the (f) linear PGE and (g) circular PGE in an
I-breaking WSM, where ultrafast ballistic carriers are excited with either
linearly or circularly polarized light, respectively. (h) Electronic
bandstructure of an inversion and mirror symmetry-breaking WSM undergoing the
CPGE. Nodes of opposite chirality are offset in energy. Circularly polarized
light of $2|\epsilon_{1}|<\hbar\omega<2|\epsilon_{2}|$ only excites momentum-
matched carriers and Pauli blocking forbids contribution from one node,
creating a net photocurrent [33].

Finally, the application of a momentum bias instead of an external magnetic
field can be used to achieve nonreciprocal optical energy transfer [34, 35, 5,
36, 37, 38]. A practical example of a static momentum bias is plasmon Fizeau
drag, where a DC current bias imparts a Doppler frequency shift on co- and
counter-propagating SPP modes [39, 40, 41, 29, 42, 43, 44]. Current flow
breaks the Lorentz reciprocity of a material by altering the optical
conductivity such that
$\sigma(\mathbf{q},\omega)\neq\sigma(-\mathbf{q},\omega)$, where $\mathbf{q}$
is the wavevector and $\omega$ is the angular frequency. This approach is
advantageous because it can be minimally invasive to on-chip device
architectures, simply requiring a DC bias.

However, the key requirement for achieving noticeable plasmon Fizeau drag is a
large carrier velocity as compared to the group velocity of the SPPs,
$v_{g}=d\omega/dq$ [43]. For this reason, both theoretical [41, 39, 40, 42,
21] and experimental [43, 45] demonstrations of appreciable plasmon Fizeau
drag have been limited to graphene, Fig. 1(a), because it achieves
exceptionally high drift carrier velocities under a DC bias [31, 32, 46]. All
other theoretical studies consider materials with unrealistic, hypothetically
relativistic charge carrier velocities [29], when in reality carrier
velocities typically saturate at much lower velocities in noble metals, Fig.
1(b), [43, 47, 29]. For example, in a copper wire with a 1 mm2 cross-section
under a 20 A current, charge carriers only reach a velocity of $1.6\times
10^{-3}$ m/s [48].

Here, we propose to use bulk WSM and Dirac semimetal (DSM) materials as a new
platform to achieve appreciable plasmon Fizeau drag-induced nonreciprocity,
Fig. 1(c). This is motivated by the exceptionally high charge carrier mobility
[49, 50, 26, 51], Fermi velocity [25], and drift current density [52, 30, 53]
reported in some WSMs and DSMs. For example, the Fermi velocity in the DSM
Cd3As2 has been shown to be equal to or even larger than that of graphene [27,
54, 55, 30]. Under a DC bias, the drift velocity, $v_{d}$, of carriers in
Cd3As2 has been measured to reach $\sim 60\%$ of the Fermi velocity [52, 30,
53], an exceptionally high value, comparable to that seen in the graphene
Fizeau drag experiments of Ref. [43]. These velocities are plotted in the top
half of Fig. 1(e), alongside those of noble metals, which are orders of
magnitude smaller.

Here we present a new nonlocal optical model of plasmon Fizeau drag in 3D
Dirac and Weyl materials and propose this effect as a feasible means to induce
nonreciprocity. First, we present the quasi-classical description of the
plasmon Fizeau drag optical response of 3D Dirac and Weyl semimetals by
deriving the Fizeau drag-modified material polarizability and optical
conductivity. In support of this we report the semiclassical, nonlocal
derivation of the polarizability of a 3D Dirac material. We then present
modeling results of plasmon Fizeau drag in the DC-biased DSM Cd3As2. We show
that the unique blend of Fizeau drag, nonlocality, and loss in this material
enables windows of pseudo-unidirectional transport, which cannot be predicted
by local models of DSM/WSM conductivities. We show that, like in the case of
magneto-optics and TRS-breaking WSMs, SPP transport can never truly be
unidirectional in the face of losses and nonlocality [56, 57], however, we
introduce new quantitative FOMs to evaluate and rank these systems for their
potential for pseudo-unidirectional SPP transport.

Further, we propose an alternative method to enable large Fizeau drag in WSMs
and DSMs through the excitement of ultrafast carriers via the photogalvanic
effect (PGE) in I-breaking WSMs and strained DSMs, Fig. 1(f-h).

##  II Results

###  II.1  Quasi-classical description of plasmon Fizeau drag in Dirac and
Weyl semimetals via drift current

To model the effect of plasmon Fizeau drag on the optical response of bulk WSM
and DSM materials, we derive a quasi-classical expression for their current-
bias-modified nonlocal dynamical polarizability,
$\Pi(\mathbf{q},\omega)\rightarrow\Pi^{u}(\mathbf{q},\omega)$, and the
corresponding optical conductivity,
$\sigma(\mathbf{q},\omega)\rightarrow\sigma^{u}(\mathbf{q},\omega)$, where the
$u$-superscript represents the current-modified form. To the best of our
knowledge, this is the first time the plasmon Fizeau drag effect in 3D
WSM/DSMs is studied. As we demonstrate and discuss in detail below, properly
accounting for both nonlocality and dissipative loss in 3D WSM/DSM materials
is crucial for identifying the conditions for unidirectional SPP transport.

Figure 2: The longitudinal dynamical polarizability function, $\Pi(q,\omega)$,
calculated via the RPA model [58, 59] (a,d) and via the semiclassical,
nonlocal (SC,NL) model introduced in this work (b,e). In (b,e),
$\Pi^{SC,NL}(q,\omega)$ is plotted in the $2\mu-\hbar v_{F}q-\hbar\omega>0$
domain (i.e., in the triangular area under the black dashed line), and the RPA
solution is shown for comparison for $2\mu-\hbar v_{F}q-\hbar\omega<0$, above
the dashed line. In (c,f), the difference between the RPA and semiclassical
$\Pi(q,\omega)$ is plotted. All plots are in units of
$\mu^{2}/\hbar^{3}v_{F}^{3}$.

The regular, unbiased longitudinal dynamical polarizability,
$\Pi(\bf{q},\omega)$ (also called the density-density response
$\Pi_{\rho,\rho}(\bf{q},\omega)$, however we drop the $\rho,\rho$ subscript),
and resulting dielectric function, $\varepsilon(\mathbf{q},\omega)$, of doped
WSM/DSMs have been derived in the literature via the first principles linear
response theory in the random phase approximation (RPA) [58, 59, 60, 61]:

$\displaystyle\mbox{Re}\Pi^{RPA}({{q}},\omega)$
$\displaystyle=-\frac{gq^{2}}{24\pi^{2}\hbar
v_{F}}\ln{\frac{4v_{F}^{2}q_{max}^{2}}{|v_{F}^{2}q^{2}-\omega^{2}|}}-\frac{gq^{2}}{8\pi^{2}\hbar
v_{F}}[C(q,\omega)+D(q,\omega)],$ (1)
$\displaystyle\mbox{Im}\Pi^{RPA}({{q}},\omega)$
$\displaystyle=-\frac{gq^{2}}{24\pi\hbar v_{F}}\Theta(\omega-
v_{F}q)-\frac{g}{8\pi\hbar
v_{F}}\begin{cases}\zeta(q,\omega)-\zeta(q,-\omega),&\text{1A}\\\
\zeta(q,\omega),&\text{2A}\\\ -\frac{q^{2}}{3},&\text{1B}\\\
-\zeta(-q,-\omega),&\text{2B}\\\ 0,&\text{3A,3B}\end{cases}$

where $\mathbf{q}=q\mathbf{\hat{z}}$, $q_{max}=\epsilon_{max}/\hbar v_{F}$ is
the cutoff wavevector corresponding to UV energy cutoff $\epsilon_{max}$, and
$g$ is the degeneracy (for WSMs, $g=2g_{W}$, where $g_{W}$ is the number of
pairs of Weyl nodes). The domains in Eq. 1 are defined as

$\displaystyle\mbox{1A}:{}{}$ $\displaystyle 0<\omega<v_{F}q$ and
$\displaystyle 2\mu-\hbar v_{F}q-\hbar\omega>0,$ (2)
$\displaystyle\mbox{2A}:{}{}$ $\displaystyle 0<\omega<v_{F}q$ and
$\displaystyle\pm 2\mu\mp\hbar v_{F}q+\hbar\omega>0,$
$\displaystyle\mbox{3A}:{}{}$ $\displaystyle 0<\omega<v_{F}q$ and
$\displaystyle 2\mu-\hbar v_{F}q+\hbar\omega<0,$ $\displaystyle\mbox{1B}:{}{}$
$\displaystyle 0<v_{F}q<\omega$ and $\displaystyle 2\mu-\hbar
v_{F}q-\hbar\omega>0,$ $\displaystyle\mbox{2B}:{}{}$ $\displaystyle
0<v_{F}q<\omega$ and $\displaystyle\mp 2\mu\pm\hbar\omega+\hbar v_{F}q>0,$
$\displaystyle\mbox{3B}:{}{}$ $\displaystyle 0<v_{F}q<\omega$ and
$\displaystyle 2\mu+\hbar v_{F}q-\hbar\omega<0,$

and the defined functions are

$\displaystyle\zeta(q,\omega)$
$\displaystyle=\frac{1}{12\hbar^{3}v_{f}^{3}q}[(2\mu+\hbar\omega)^{3}-3\hbar^{2}v_{F}^{2}q^{2}(2\mu+\hbar\omega)+2\hbar^{3}v_{F}^{3}q^{3}],$
(3) $\displaystyle C(q,\omega)$
$\displaystyle=\frac{8\mu^{2}}{3\hbar^{2}v_{F}^{2}q^{2}}-\frac{\zeta(q,\omega)H(q,\omega)}{q^{2}}-\frac{\zeta(-q,\omega)H(-q,\omega)}{q^{2}},$
$\displaystyle D(q,\omega)$
$\displaystyle=C(q,-\omega)-\frac{8\mu^{2}}{3\hbar^{2}v_{F}^{2}q^{2}},~{}~{}H(q,\omega)=\ln{\frac{|2\mu+\hbar\omega-\hbar
v_{F}q|}{|\hbar v_{F}q-\hbar\omega|}},$

where $\mu$ is the chemical potential. Note that chemical potential is
temperature dependent, $\mu(\epsilon_{F},T)$, and at zero temperature is equal
to the Fermi energy $\epsilon_{F}$.

RPA is a nonlocal correlation energy functional, and thus the RPA-derived
polarizability captures nonlocal optical response driven by both inter-band
and intra-band electronic transitions. However, RPA may come with a high
computational cost and may suffer from convergence issues and instabilities
[62]. Instead, well-known, semiclassical Drude models are often used to
describe longitudinal polarizability. In the local approximation at zero
temperature, in the absence of dissipative losses, and at frequencies below
the onset of intra-band transitions, this has been shown to be [61]:

$\displaystyle\Pi^{SC,L}({q},\omega)$
$\displaystyle=\frac{1}{3}gN(\mu)\frac{q^{2}v_{F}^{2}}{\omega^{2}},$ (4)

where $N(\epsilon)={\epsilon^{2}}/{2\pi^{2}\hbar^{3}v_{F}^{3}}$ is the
electron density of states for a WSM/DSM, $\epsilon=\hbar\omega$ is energy,
and the $SC,L$-superscript indicates semiclassical and local.

Here, aiming to derive a computationally-efficient yet accurate approximation
for the WSM/DSM polarizability and dielectric function, we follow a
semiclassical framework to first derive the expressions for the unbiased
$\Pi({q},\omega)$ and $\sigma({q},\omega)$ accounting for nonlocality. To the
best of our knowledge, all semiclassical optical models for 3D WSM/DSMs in the
literature are in the local approximation, as in Eq. 4 [61]. As we will show,
it is essential to use a nonlocal optical model to model Fizeau drag in
WSM/DSMs because transport in Dirac materials is highly nonlocal [63] and
because the Fizeau drag induces additional, compounding nonlocality on the
material [44]. Thus, we provide here the first reporting of the semiclassical,
nonlocal optical conductivity/permittivity for WSM/DSMs, derived via a
semiclassical Boltzmann equation formalism. We find this polarizability to be

$\displaystyle\Pi^{SC,NL}({{q}},\omega)$
$\displaystyle=-gN(\mu)\left(\frac{\omega}{qv_{F}}\frac{1}{2}\ln{\frac{\omega-
qv_{F}}{\omega+qv_{F}}}+1\right),$ (5)

where the $SC,NL$-superscript indicates semiclassical and nonlocal. We provide
the detailed derivations for $\Pi^{SC,NL}({q},\omega)$,
$\sigma^{SC,NL}({q},\omega)$ and $\epsilon^{SC,NL}({q},\omega)$ in
Supplementary Note I, and the real and imaginary parts of Eq. 5 are expressed
explicitly in Supplementary Eq. S11. The $SC,NL$-superscript is dropped for
conciseness in the remaining text.

In the semiclassical regime, it is assumed that the energy scales of
$\epsilon=\hbar\omega$ and $v_{F}q$ are much less than $\mu$. To evaluate the
range of applicability of Eq. 5, we compare its prediction of the
polarizability function to the corresponding RPA result over the $2\mu-\hbar
v_{F}q-\hbar\omega>0$ domain in Fig. 2. The comparison of the two models shows
their reasonable agreement. As an additional validation of the new model, the
unbiased SPP dispersion is plotted for the Cd3As2 case in Supplementary Fig.
S3, also matching well with the results of the RPA model. It should be noted
that the RPA and semiclassical expressions for the polarizability (i.e., Eqs.
1, 4, and 5) are valid in the low-temperature regime when $kT\ll\epsilon_{F}$.

We then extend the derivation to obtain the current-biased expressions
$\Pi^{u}({q},\omega)$ and $\sigma^{u}({q},\omega)$, which account for a
current-modified charge density distribution in the material and enable
estimations of the Fizeau drag effect on the SPP modes propagation. This
derivation is outlined in detail in Supplementary Note II. Similarly to the
unbiased case, we follow the Boltzmann transport approach, however, the
equilibrium Fermi distribution is now skewed by the current bias, Fig. 1(d),
becoming [43, 39]:

$f^{u}(\epsilon)=\frac{1}{e^{(\epsilon(\mathbf{p})-{\mathbf{u\cdot
p}}-\mu^{u})/k_{B}T}}.$ (6)

Here, $\mathbf{u}$ is the drift velocity of the quasiparticles, $\mathbf{p}$
is the quasiparticle momentum, $\epsilon(\mathbf{p})$ is energy, and $\mu^{u}$
is a modified chemical potential. The effect of Eq. 6 on the material
polarizability and optical conductivity can be evaluated using a Lorentz
transformation. However, due to their linearly dispersing electron bands,
$\epsilon({\bf p})=\mbox{sgn}(\mu)v_{F}|{\bf p}|$, Weyl and Dirac fermions are
characterized by a quasi-Lorentz invariance, where the Fermi velocity $v_{F}$
replaces the speed of light in the usual Lorentz factor,
$\gamma=(1-u^{2}/v_{F}^{2})^{-1/2}$, as has been discussed for graphene [39,
64]. Thus, in a current-biased regime, the Doppler effect is expected to
follow a quasi-Lorentzian transform (QLT). For ${\bf u}\parallel{\bf
q}\parallel\mathbf{\hat{z}}$, the electromagnetic and electron frequency and
momentum are transformed under the QLT as

$\displaystyle\omega_{0}=\gamma(\omega-
uq^{z}),~{}q_{0}^{z}=\gamma(q^{z}-\frac{u}{v_{F}^{2}}\omega),$ (7)
$\displaystyle q_{0}^{y}=q^{y},~{}q_{0}^{x}=q^{x}$
$\displaystyle\epsilon_{0}(p)=\gamma\left(\epsilon(p)-up^{z}\right),~{}p_{0}^{z}=\gamma\left(p^{z}-\frac{u}{v_{F}^{2}}\epsilon(p)\right)$
, (8) $\displaystyle p_{0}^{y}=p^{y},~{}p_{0}^{x}=p^{x}$

where the $0-$subscript designates the moving frame. Under the QLT, the
perturbed carrier density is transformed as

$f^{u}(\epsilon)=\frac{1}{e^{(\epsilon-
up^{z}-\mu^{u})/k_{B}T}+1}=\frac{1}{e^{(\epsilon_{0}-\mu_{0})/\gamma
k_{B}T}+1},$ (9)

where we have introduced the transformed chemical potential
$\mu_{0}=\gamma\mu^{u}$. To conserve carrier density the chemical potential
must transform as $\mu_{0}=\mu/\gamma^{1/3}$ (Supplementary Eq. S30).
Following the derivation detailed in Supplementary Note II, the longitudinal
polarizability in a current carrying state is found to be

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=-gN(\mu_{0})\frac{q^{2}}{q_{0}^{2}}\left(\frac{\omega_{0}}{2q_{0}v_{F}}\ln\left(\frac{\omega_{0}-q_{0}v_{F}}{\omega_{0}+q_{0}v_{F}}\right)+1\right).$
(10)

Transforming Eq. 10 according to
${\sigma^{u}}({q},\omega)=(ie^{2}\omega/q^{2})\Pi^{u}({q},\omega)~{}$[65], the
current-biased, longitudinal optical conductivity is transformed via the
relation

$\displaystyle\sigma^{u}(\omega,{q},\mu)$
$\displaystyle=\frac{\omega}{\omega_{0}}\sigma_{zz}(\omega_{0},{q_{0}},\mu_{0}).$
(11)

From $\sigma^{u}$, the impact of DC bias Fizeau drag on SPPs at 3D WSM/DSM
surfaces can be modeled. Note that because we are considering TRS-preserving
WSMs for Fizeau drag, we do not solve for or consider the Fizeau drag impact
on the anomalous hall effect or $\sigma_{AHE}$ in this work.

###  II.2 Photogalvanic Effect enabled plasmon Fizeau Drag

While the exceptional mobility and current density of some 3D WSM/DSMs already
make them good candidates for Fizeau drag via DC bias, we propose an
additional Fizeau drag mechanism via the PGE, Fig. 1(f-g). The linear PGE
(LPGE), also called shift current, is manifested as the shift in space of a
charge carrier during the inter-band photoexcitation process under
illumination by linearly polarized light, stemming from the I-symmetry
breaking of the material [66]. Berry flux fields contribute to shift current,
making WSMs attractive material candidates for observing this effect [67, 68,
69]. In the circular PGE (CPGE), illumination from circularly polarized light
selectively excites charge carriers with preferred linear momenta, Fig. 1(h),
supporting a net photocurrent in certain symmetry groups of I-symmetry-
breaking WSM or strained DSM materials [70, 71, 72]. Importantly, the excited
ballistic carriers propagate with velocity $v_{F}$ over their mean free path
[66].

By taking advantage of the ultrafast ballistic carriers continuously generated
via the PGE mechanism, we can bypass the carrier velocity limits achievable
with drift current via DC bias. We propose that the PGE photoexcited carriers
can impart Fizeau drag on the electron plasma, thus providing an external
mechanism to control nonreciprocal behavior by optical pumping. The $v_{F}$ is
inherently high in many WSM and DSM materials, while the saturation velocity
in the DC-biased regime is limited by material synthesis, Fig 1(e). This
mechanism could thus allow observations of strong Fizeau drag in many more
material candidates than what is achievable via DC bias.

As shown in Fig. 1(g), the CPGE current mechanism is inherently different from
DC bias, Fig. 1(d), in the electron bandstructure picture, resulting in
different equilibrium charge distributions. Thus, the optical model derived
using Eq. 6 is inherently only applicable to the DC-biased case; a different
treatment of the equilibrium carrier distribution should be used for the PGE
case which accounts for the relative excited band populations. However,
related THz-pumping experiments in topological insulators (TIs) have shown
that the DC bias modeling done here may, in fact, provide a good approximation
for the PGE-induced Fizeau drag when modified to consider pumping dynamics.

In THz pumping experiments of the TI Bi2Te3, ultrafast and strong fields were
found to accelerate the Dirac surface states, in a phenomenon called light-
field-driven currents [73], which is distinct from, but related to the PGE
[74]. Under a semiclassical approximation, the current density was $j\approx-
ev_{f}n\Delta k(t)/k_{F}$, where $\Delta k(t)$ was the shift in the electron
density observed in the direction of the current via angle-resolved
photoemission spectroscopy (ARPES). The time-dependent electron distribution
of the Dirac cone was approximated, assuming a small relaxation time
$\tau_{R}$, as

$\displaystyle f(\epsilon,t)$
$\displaystyle\approx\frac{1}{e^{(\epsilon(\mathbf{p})-\mathbf{v_{F}}\cdot\Delta\mathbf{p}(t)-\mu)/k_{B}T}+1},$
(12)

where $\Delta\mathbf{p}(t)=\hbar\Delta\mathbf{k}(t)$, is a function of
$\tau_{R}$ and the pumping field $E(t)$. This semiclassical description
matched the experimental results for Bi2Te3 THz pumping experiments, and
$\Delta\mathbf{k}(t)$ was directly proportional to the temporal current
density measurement. Compared to Eq. 6, we see that Eq. 12 is equivalent with
the exception that the $\Delta\mathbf{k}$ terms have different physical
origins and that the latter is time dependent.

Both the DC-biased, Eq. 6, and PGE, Eq. 12, cases can be considered as cases
where the chemical potential, $\mu$, becomes band-dependent

$\displaystyle\mu(\epsilon_{F},T)$
$\displaystyle\rightarrow\mu(\epsilon_{F},T,\delta n).$ (13)

Under DC biasing, $\mu(\epsilon_{F},T,\delta
n)\approx\mu^{u}(\epsilon_{F},T)+\mathbf{u}\cdot\mathbf{p}$ (Eq. 9), and under
photoexcitation $\mu(\epsilon_{F},T,\delta
n)\approx\mu(\epsilon_{F},T)+\mathbf{v}_{F}\cdot\Delta\mathbf{p}(t)$. In both
cases, the electron density is modified by an external perturbation $\delta
n$: in the case of drift current $\delta n$ is due to a voltage drop across
the material, while in the case of optical pumping, $\delta n$ is a result of
photoexcitation [75].

Similar ARPES experiments could be conducted for the transient electron
distribution in WSM/DSMs under the PGE. From these experiments, a similar
semiclassical treatment of the PGE pumping in WSM could be performed. Such
results for $\Delta\mathbf{k}(t)$ of a 3D Dirac or Weyl material under the PGE
could extend the analysis performed here for the DC drift-biased case to the
PGE pumping case.

It is important to acknowledge that - unlike the DC-biased case - the PGE
Fizeau drag approach requires an external pumping field, but in principle can
be activated without the electrodes, which simplifies the on-chip design and
fabrication requirements. However, under prolonged pumping, a generated space-
charge field can lead to so-called ’optical damage’ [76], which can be
alleviated by adding the electrodes similar to the DC-biased case. Overall,
the proposed alternative way to achieve the Fizeau drag holds interest in
studying carrier dynamics and interactions in a wide range of Weyl and Dirac
materials and can be used to develop new techniques to achieve and models to
describe the alteration of symmetry at ultrafast timescales. This mechanism
could add to the toolbox of THz-driven controls of quantum materials and
studies of many-body dynamics [77].

###  II.3 Fizeau-drag SPP dispersion in 3D Weyl and Dirac semimetals

We model plasmon Fizeau drag under drift current bias using the transformed
nonlocal Drude model in Eq. 10. We look at the example case of Cd3As2, a 3D
DSM, which has ultrahigh experimentally-measured carrier mobility,
$\mu=9\times 10^{6}$ cm2V-1s-1and Fermi velocity, $v_{F}=9.3-1.5\times 10^{5}$
m s-1 [27]. The current density has been reported to exceed $5~{}\mbox{A/mm}$
for a carrier density of $5\times 10^{12}\mbox{cm}^{-2}$, corresponding to an
exceptionally high drift velocity of $u=6.3\times 10^{5}$ ms-1 $\approx
0.6v_{F}$ [52]. Epitaxially grown thin films of Cd3As2 have also demonstrated
high mobility and $v_{F}$ [78, 79].

Figure 3: (a) Dispersion curves for the SPP modes supported on the interface
(b) between a bulk DSM Cd3As2 material and air, which are calculated in the
lossless regime with and without drift bias ($\epsilon_{F}=0.15$ eV,
$v_{F}=1\times 10^{6}$ ms-1, $\epsilon_{\infty}=13$, $\epsilon_{b}=12$, $g=4$
[61]). The upward bending of the SPP modes at high $q$ values is a
characteristic signature of nonlocality. (c) The nonreciprocity of the
lossless SPP modes at $\omega=0.369\epsilon_{F}$, corresponding to the dashed
line in (a), is represented as the Fizeau shift,
$(\lambda_{fwd}-\lambda_{bwd})/\lambda_{0}$, plotted versus the bias drift
velocity. (d) The co- and counter-propagating nonreciprocal SPP modes compared
for a larger range of drift bias velocities. (e) The calculated 1D-PDOS values
of both modes reveal that the counter-propagating mode increasingly dominates
at lower energies as the drift velocity increases.

We model the system assuming that drift current is in the longitudinal
direction, and the z-axis as illustrated in Fig. 3(b). The longitudinal
optical conductivity, $\sigma_{L}=\sigma_{zz}$, is transformed according to
the QLT in Eq. 11. The transverse optical conductivity,
$\sigma_{T}=\sigma_{yy}$, is expected to exhibit a weaker transformation,
which is neglected here for simplicity, $\sigma_{T}^{u}\approx\sigma_{T}$ [39,
80, 44]. The optical conductivity in the intra-band model is transformed to
permittivity according to
$\varepsilon_{ij}(q,\omega)=\varepsilon_{\infty}(\omega)+i{\sigma_{ij}(q,\omega)}/{\varepsilon_{0}\omega}$
[81, 61], and the SPP eigenmodes are solved for with the dispersion relation
[44]

$q=\pm\frac{\omega}{c}\sqrt{\varepsilon_{1}\frac{\varepsilon^{u}_{T}(\varepsilon_{L}^{u}-\varepsilon_{1})}{\varepsilon_{L}^{u}\varepsilon_{T}^{u}-\varepsilon_{1}^{2}}},$
(14)

where $\varepsilon_{1}$ describes the permittivity of the medium above the
WSM/DSM.

First, we consider the lossless case for carriers with drift velocity,
$v_{d}=u$, varying between $10\%$ and $50\%$ of $v_{F}$, Fig. 3(a,d). The
Fizeau drag introduces a second bulk mode to the dispersion characteristics of
this photonic structure, similar to the appearance of the lower and upper bulk
modes in magnetized plasmas and TRS-breaking WSMs [57]. With increased drift
velocity, both, the dispersion branch corresponding to the counter-propagating
surface mode (i.e., the mode with the $-q$-wavevector direction in this
configuration) and the lower bulk mode branch are progressively red-shifted.
Such Fizeau-drag-induced Doppler red-shifts have also been predicted for SPPs
on metals [44] and demonstrated experimentally for graphene SPPs [43].
Finally, similarly to the DC-biased SPP modes on metal interfaces, an
inflection point appears in the counter-propagating mode dispersion curve, at
which point the group velocity of the SPP is reversed [44]. This inflection
point has however not been observed for the DC-biased SPP modes in 2D Dirac
materials (graphene) [43].

In prior works, the Fizeau shift, $(\lambda_{fwd}-\lambda_{bwd})/\lambda_{0}$,
has been used as the figure of merit (FOM) to quantify nonreciprocity of the
SPP mode in the lossless case [45, 43]. This FOM is plotted in Fig. 3(c) as a
function of the DC bias drift carrier velocity [43], where $\lambda_{0}$ is
the SPP wavelength without bias. We evaluate the Fizeau shift at
$\omega=0.369\epsilon_{F}=13.4$ THz, which corresponds to unbiased
$\lambda_{0}=360$ nm in Fig. 3(c). These results are qualitatively comparable
to the Fizeau shift measured in graphene at the same drift velocities [43].
However, by comparing the dispersion plots for a 3D DSM (Fig. 3) and graphene
(Supplementary Fig. S6), we can conclude that this FOM does not adequately
estimate the effect of the nonreciprocity and does not allow for proper
comparison between different material systems and external biases.

While both dispersion curves in Fig. 3 and Supplementary Fig. S6 exhibit
significant nonreciprocity, the capability of 2D and 3D Dirac materials to
achieve a unidirectional SPP transport is hindered by nonlocality in different
ways. In both cases, true unidirectional transport is not possible due to
nonlocality, and the counter-propagating mode experiences red shift. However,
differently from the 2D graphene case, the inflection point observed in the 3D
DSM $-q$ SPP branch opens up a narrow window of a pseudo-unidirectional SPP
transport in the counter-propagation direction, evident for $u=0.3v_{F}$ and
$u=0.5v_{F}$ in Fig. 3(d).

To properly evaluate the nonreciprocity and its effect on achieving highly-
asymmetrical SPP transport within some frequency windows, we propose to use a
different FOM by calculating and comparing the density of photon states
available for the SPP modes co-propagating and counter-propagating with the
bias current. This new universal FOM - The 1D photonic density of states
(1D-PDOS) - is calculated by counting the available states in the momentum
(q-vector) space corresponding to a narrow frequency window sliding along the
frequency axis:

$\mbox{1D-PDOS}(\omega)=\int_{\omega-\delta\omega}^{\omega+\delta\omega}q(\omega)d\omega.$
(15)

1D-PDOS is calculated numerically by separately integrating over the
dispersion curves in the forward and reverse directions. The results for
Fizeau drag with increasing bias velocity plotted in Fig. 3(d-e) show that a
frequency window can be opened where the counter-propagating mode (dashed
lines) exhibits a much higher 1D-PDOS (up to $\sim$15 times for 50% $v_{F}$)
than the co-propagating one (solid lines), effectively allowing a pseudo-
unidirectional SPP transport. This circumstance differs notably from the
prediction of truly unidirectional SPP transport through the co-propagating
SPP mode, a prediction achieved when invoking the non-realistic local Drude
mode (see Supplementary Fig. S3). In the nonlocal case shown in Fig. 3, both,
co- and counter-propagating modes are allowed within the pseudo-unidirectional
transport window, but the amount of energy that can be back-scattered to the
co-propagating one is severely limited by its very low DOS.

Finally, to get the full physical picture of nonreciprocal propagation of SPP
modes under current bias in 3D DSM and WSM materials, dissipative losses
should be included in the analysis together with the nonlocality.

####  II.3.1 Combined effect of dissipation and nonlocality on nonreciprocal
SPP transport

Figure 4:  Impact of finite damping on nonlocal SPP modes under a DC bias. (a)
For a scattering period of $\tau=2.1\times 10^{-10}$s and a DC bias of
$u=0.5v_{F}$, loss causes band-bending of the counter-propagating,
$-q$-wavevector mode (yellow) yet does not impact the co-propagating,
$+q$-wavevector mode (blue). The counter-propagating mode is also severely
damped, as evident from the much larger magnitude of its imaginary component
of the wavevector (compare blue and yellow dashed lines). (b) The
corresponding 1D-PDOS values and (c) propagation lengths for the co- and
counter-propagating modes show that the co-propagating mode dominates in both
1D-PDOS and propagation length at energies above 0.336$\epsilon_{F}$. In (d-g)
the evolution of the SPP dispersion is illustrated as $\tau$ is progressively
decreased, with the complementary $\mbox{Re}(q)$ and $\mbox{Im}(q)$ components
plotted as solid and dashed lines, respectively. The solutions corresponding
to the lossless case are plotted in dotted grey. In (e,f) the co-propagating
mode is preserved and underdamped, while the counter-propagating mode
experiences increased band-bending and overdamping. In (g), a damping
threshold is surpassed and the co-propagating mode also back-bends into the
bulk mode. The same parameters for Cd3As2 are used as in Fig. 3.

Dissipation is an inherent property of any plasmonic media [82] and the
impacts of finite damping on the system must be considered. Dissipative losses
cause folding of the SPP dispersion curves away from their asymptotic behavior
and thus erase the nonreciprocal modes at large wavevectors. However, in some
cases, nonlocality can open frequency windows where the SPP transport remains
bidirectional but one of the modes is significantly more overdamped than the
other [56].

To solve for the SPP modes in the lossy case we search for solutions across a
complex-$q$ plane, as described in Supplementary Note III. Owing to the
ultrahigh mobility of Cd3As2, the electron transport lifetime is exceptionally
high: $\tau=2.1\times 10^{-10}$s [27]. When damping is accounted for in the
model, as plotted in Fig. 4(a) for $u=0.5v_{F}$ and $\tau=2.1\times
10^{-10}$s, the counter-propagating mode (yellow) experiences the usual loss-
induced back-bending behavior. At the same time, the co-propagating mode
(blue) does not exhibit back-bending and acquires much lower $\mbox{Im}(q)$
compared to the counter-propagating mode. Now, in addition to the 1D-PDOS,
Fig. 4(b), in the lossy case we also evaluate and compare the propagation
lengths of both SPP modes. The propagation length is defined here as the
distance over which the co- or counter-propagating SPP mode power decays by a
factor of $1/e$, $L_{\pm}=1/(2\mbox{Im}(q))$ [83]. Fig. 4(c) shows that in the
designated frequency window, the counter-propagating mode is significantly
overdamped, $\mbox{Im}(q)>\mbox{Re}(q)$, while the co-propagating mode is
underdamped, $\mbox{Im}(q)<\mbox{Re}(q)$. This results in almost zero
propagation length of the counter-propagating mode as compared to the co-
propagating mode, whose propagation length can span from tens to hundreds of
microns. However, at lower frequencies, the counter-propagating mode exhibits
a much higher 1D-PDOS, which may translate into a high level of propagation
loss if the forward mode scatters on any material impurities or surface
imperfections.

Figure 5: The SPP dispersion and corresponding 1D-PDOS for the (a) unbiased
and increasing drift current bias of (b) $u=0.3v_{F}$, (c) $u=0.5v_{F}$, and
(d) $u=0.7v_{F}$ in the 3D DSM Cd3As2. The dashed black line in the 1D-PDOS
represents the energy when the bulk mode begins. The propagation length is
called out at a specific energy where both the relative propagation length and
1D-PDOS of the co-propagating mode (blue) dominate over the corresponding
parameters of the counter-propagating mode (yellow). The same parameters for
Cd3As2 are used as in Fig. 3 with $\tau=2.1\times 10^{-11}$s.

To probe the effect of progressively increasing dissipative losses on the SPP
transport, we model the dispersion across the scattering period range
$\tau=2.1\times 10^{-10}-2.1\times 10^{-12}s$ [49]. These data are plotted in
Fig. 4(d-g) and show that Fizeau drag increases the nonlocality in the co-
propagating mode, while it makes the counter-propagating mode act more local,
resulting in the former being underdamped while the latter is overdamped. At
high enough scattering rates, Fig 4(g), the co-propagating mode eventually
experiences band-bending, and both modes become overdamped. In the shaded
regions in (e-f), while the counter-propagating mode is still an available
channel for the optical energy dissipation via absorption, its overdamped
nature translates to both low 1D-PDOS values and negligible propagation
lengths. This effectively prevents SPP backpropagation, opening a window of
pseudo-unidirectional surface mode transport despite the nonlocality. This is
a key result of our work, showing that Fizeau drag in 3D WSM/DSMs may unlock
practical windows of pseudo-unidirectional SPP transport under real-world
nonlocal and lossy conditions.

Figure 6:  Comparison of the SPP dispersion relations and resulting 1D
photonic density of states (1D-PDOS) in the nonlocal, lossy regime at the
interface of a (a-b) Fizeau drag current-biased DSM, (c-d) magneto-optic (data
from [56]), (e-f) TRS-breaking WSM. The dashed black line in the 1D-PDOS
represents the energy when the bulk mode begins. The DC-biased DSM/WSM is
modeled with the same parameters as Fig. 3, with $\tau=2.1\times 10^{-11}$s
and $u=0.3v_{F}$. The TRS-breaking WSM is modeled with the commonly used
parameters of $\epsilon_{F}=0.15$ eV, $v_{F}=0.83\times 10^{5}$ ms-1,
$\epsilon_{\infty}=13$, $\epsilon_{b}=6.2$, $g=2$, a Weyl node separation of
$b=2\times 10^{9}$, and $\tau=2.1\times 10^{-11}$s [13, 15] against an air
interface. The magneto-optic SPP was modeled in [56] using a silicon-InSb
interface. The silicon is modeled with $\varepsilon=11.68$ and InSb was
characterized by $\varepsilon_{\infty}=15.6$, $\omega_{p}=2\times 10^{12}$ Hz,
$\tau=1/0.025\omega_{p}=3.2\times 10^{-12}$s, and an applied magnetic field of
$B=0.2$T. Plots (b,d,f) have the same $q$-axes and and the $\omega$-axes have
scales of 1 THz (b,d) and 1.5 THz (f) to show a relatively clear comparison in
magnitude of nonreciprocity. Landau damping occurs for
$q_{d}>\omega_{P}/v_{F}$ which is outside of the bounds of these plots for
each case [82].

Similar predictions have previously been made for nonlocal lossy SPP modes
supported by interfaces of magneto-optic materials and time-reversal-symmetry
breaking WSMs [57, 56]. To evaluate the advantages and pitfalls of different
mechanisms of inducing nonreciprocity in 3D materials, in Fig. 6 we compare
dispersion and transport characteristics of SPPs in WSM/DSMs under Fizeau drag
(a,b) with those of SPPs in magneto-optic materials under external magnetic
bias (c,d) and in TRS-symmetry breaking WSMs (e,f) when loss and nonlocality
are taken into account. The current-biased WSM/DSM is modeled with the same
parameters as in Fig. 3-5 with $u=0.3v_{F}$ and $\tau=2.1\times 10^{-11}$s.

In comparison to the WSM/DSM Fizeau drag results, when nonlocality and loss
are accounted for in the magneto-optic InSb, Fig. 6(c), results from Ref. [56]
show that the forward and reverse propagating modes (1) both exhibit upward
bending behavior and (2) are both underdamped, $\mbox{Im}(q)<\mbox{Re}(q)$,
meaning both modes propagate. This result is replicated in Fig. 6(d). The
1D-PDOS plot shows that an energy window exists where the backward (yellow)
mode dominates in the number of available states, $\omega\approx 1.4-1.6$ THz,
opening a broad window of pseudo-unidirectional transport, demonstrated by the
large 1D-PDOS${}_{-}/$1D-PDOS+ ratio. When the two modes overlap at higher
$\omega$ they are in competition, with the same relative propagation length.

Next, we model the SPPs at a TRS-breaking WSM interface, Fig. 6(e-f), with the
material parameters from Ref. [15] and $\tau=2.1\times 10^{-11}$s. We apply
our semiclassical, nonlocal polarizability function, Eq. 5, in calculating the
diagonal components of the permittivity. Even when accounting for nonlocality,
at this magnitude of loss, the familiar back bending behavior [16] for the
$\pm q$ modes is present. The $-q$ (yellow) mode is extremely dominant in
1D-PDOS, with the $+q$ (blue) mode so much less populous that it is not
visible. However, at the peak of the $-q$ 1D-PDOS, the propagation length of
the backward mode, $L_{-}$, is relatively low. In contrast, $L_{+}$ is much
larger, but the $+q$-mode has virtually no available states in comparison to
the $-q$ one. Despite this, this situation presents the most distinct pseudo-
unidirectional SPP transport with the stark gap in 1D-PDOS between the $\pm q$
modes (with the 1D-PDOS ratio $>$3000 at 20.25 THz).

Landau damping will additionally suppress the SPP modes in Fig. 6 at
wavevectors $q_{d}>\omega_{P}/v_{F}$ [82]. For the DSM modeled under Fizeau
drag, Landau damping is predicted to occur at $q_{d}=2\times 10^{8}$ m-1,
while for the TRS-breaking WSM at $q_{d}=2.5\times 10^{9}$ m-1, and for the
magneto-topic InSb at $q_{d}=1.7\times 10^{7}$ m-1. These momentum limits are
well outside of the domains plotted in Fig. 6. Surface-induced Landau damping
will also impact the SPPs and is accounted for in the magneto-optic results
[56] but not for the DSM/WSM cases, Fig. 6(a,e). Inclusion of surface
collisions would increase the scattering rates of these systems [84], though
the effects of nonlocality may remain much more pronounced [56].

These results demonstrate the importance of accounting for nonlocality and
loss in discussions of nonreciprocal transport. Additionally, they emphasize
the importance of considering the 1D-PDOS± and propagation lengths, $L_{\pm}$,
as key figures of merit for quantifying the directionality and efficiency of
the SPP energy transport.

##  III Discussion

In summary, we have presented the theoretical framework for plasmon Fizeau
drag in 3D Dirac and Weyl materials, enabling it to join the toolbox of
mechanisms for tuning the nonreciprocity of topological materials. Importantly
for engineering applications, we have shown that the unique blend of
nonlocality and loss in the DSM Cd3As2 enables windows of pseudo-
unidirectional transport.

Further, we have introduced new FOMs for evaluating and ranking nonreciprocal
and unidirectional SPP transport: 1D-PDOS± and propagation length, $L_{\pm}$.
These FOMs allow us to compare the potential for pseudo-unidirectional
transport in Fizeau drag-biased 3D WSM/DSMs to that of magneto-optic materials
and TRS-breaking WSMs. In terms of practicality, the magneto-optic case in Fig
6(c-d) requires a modest applied magnetic field of B=0.2T and a nonreciprocal
absorber/emitter was recently demonstrated with the magneto-optic InAs under
1T [85]. In contrast, the first TRS-breaking WSM was only experimentally
demonstrated recently and, to date, the prediction of nonreciprocal optical
behavior in these materials is in the early stages of experimental
demonstration [17, 86]. The experimental demonstration of Fizeau drag in 3D
WSM/DSM will not require an externally applied magnetic field or corresponding
equipment and will also draw on a wider selection of I-breaking DSM and WSM
material candidates that may be less vulnerable to the synthesis challenges
involved with working with TRS-breaking WSM materials.

In this vein, we propose that experiments to demonstrate Fizeau drag biased 3D
WSM/DSMs be pursued. In experiment, we expect that, as was observed in
graphene [43], the DC bias may modify the higher order, nonlinear optical
conductivity components, leading to a nonlinear Fizeau shift as a function of
the DC bias field. In contrast to graphene Fizeau drag experiments, the effect
in 3D WSM/DSM should open windows of pseudo-unidirectional SPP transport,
allowing for additional experimental probes beyond measuring SPP wavelength.
In particular, the frequency and angular selectivity of THz absorbtance or
thermal emittance both scale with the 1D-PDOS values, allowing to probe
nonreciprocity via far-field spectroscopic or ellipsometric measurements,
similar to the experiments performed with magneto-optic materials under
magnetic bias [87, 88].

We have also related the PGE to Fizeau drag as a new mechanism for ultrafast
THz-driven control of quantum materials. We believe that future APRES
experiments of the CPGE and LPGE in WSM/DSMs will help to map the transient
electron distribution under the effect and enable future extensions of the
modeling presented here for the DC-biased Fizeau drag. The same material
evaluated for the drift current case, Cd3As2, is an ideal candidate for CPGE-
enabled Fizeau drag under strain [27, 28, 52]. Additionally, the Fizeau drag
effect may also be extended to surface PGE currents as well as surface
currents in TIs.

Finally, it was recently experimentally shown that optically pumping
photocurrents in type-I WSM TaAs (identified to have large $v_{F}$ in Fig.
1(e)) can break TRS in the material [89]. The degree of TRS-breaking was
directly related to the direction of photocurrent [90] and it was deduced to
be entirely electronic in origin. We posit that the electronic origin of this
TRS-breaking may be understood through the Fizeau drag mechanism presented
here.

###### Acknowledgements.

We wish to acknowledge the support of Draper Labs for funding M.G.B. through
the Draper Scholars Program. This work is supported in part by the Near Field
Radiative Heat Transfer ARO MURI (Grant No. W911NF-19-1-0279) via U. Michigan.
We thank Yoichiro Tsurimaki, Shanhui Fan, Thanh Nguyen, Simo Pajovic, and Mark
Witinski for helpful discussions.

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Supplementary Information for “Plasmon Fizeau drag in 3D Dirac and Weyl
semimetals”

Blevins and Boriskina

Table S1: Nomenclature  Acronyms |
---|---
DSM | Dirac Semimetal
LPGE | Linear Photogalvanic Effect
QLT | Quasi-Lorentz Transform
RPA | Random Phase Approximation
SC | Semiclassical
SPP | Surface Plasmon Polariton
THz | Terahertz
WSM | Weyl Semimetal
Electromagnetics |
$\omega$ | EM frequency
$\omega_{0}$ | EM frequency in moving frame
$\omega_{p}$ | EM plasma frequency
$\sigma$ | Optical conductivity
$\sigma^{u}$ | Optical conductivity for current-biased state
$\varepsilon$ | Permittivity
$\varepsilon_{\infty}$ | Background permittivity
$q$ |  $=2\pi/\lambda$, Wavevector
$q_{0}$ | Wavevector in moving frame
$q_{d}$ | Landau damping wavevector
Matter/Electrons |
$\epsilon$ | Energy
$\epsilon_{F}$ | Fermi Energy
$\gamma$ |  $=(1-u^{2}/v_{F}^{2})^{-1/2}$ Quasi-Lorentz factor
$\mu$ | Chemical potential
$\mu^{u}$ | Transformed chemical potential in current-carrying state
$\mu_{0}$ | Chemical potential in moving frame
$\phi$ | Potential
$\Pi^{u}_{\rho,\rho}$ |  $=\Pi^{u}$, Polarizability, density-density response function, under current bias
$\Pi_{\rho,\rho}$ |  $=\Pi$, Polarizability, density-density response function
$\rho_{ind}$ | Induced charge density
$k_{F}$ | Fermi momentum
$n$ | Quasiparticle density
$N(\epsilon)$ | Density of states
$n_{0}$ | Quasiparticle density in moving frame
$p$ | Quasiparticle momentum
$p_{0}$ | Quasiparticle momentum in moving frame
$p_{F}$ | Fermi momentum
$u$ | Carrier drift velocity
$v_{F}$ | Fermi velocity
$g$ | Degeneracy
$g_{W}$ |  $=g/2$ Number of pairs of Weyl nodes

##  I Supplemental Note 1: Longitudinal polarizability and optical
conductivity in DSM and WSM

###  I.1 Nonlocal Case

To enable the derivation of a Fizeau drag-modified optical conductivity for 3D
DSM/WSMs, we first derive the nonlocal Drude conductivity of the 3D DSM/WSMs
via a semiclassical Boltzmann equation formalism [91, 92]. The quasiparticle
distribution function, $f(\mathbf{p})=f^{0}(\mathbf{p})+\delta f(\mathbf{p})$,
is modeled as a combination of the equilibrium distribution,
$f^{0}(\mathbf{p})$, and a perturbation to the distribution, $\delta
f(\mathbf{p})$, via an externally applied electric field,
$\mathbf{E}=\mathbf{E_{0}}e^{-i\omega t+i\mathbf{q}\cdot\mathbf{r}}$. We will
use the collisionless regime, $(\partial f/\partial t)_{coll}\rightarrow 0$,
and account for losses afterward. The Boltzmann transport equation is

$\displaystyle\frac{\partial f}{\partial
t}+\dot{\mathbf{p}}\cdot\nabla_{p}f+\mathbf{v_{p}}\cdot\nabla_{r}f$
$\displaystyle=0.$ (S1)

Making use of $\dot{\mathbf{p}}=e\nabla_{r}\phi_{tot}(\mathbf{r},t)$ and
$\nabla_{p}f^{0}(\mathbf{p})=\nabla_{\mathbf{p}}\epsilon_{p}\frac{\partial
f^{0}(\epsilon_{p})}{\partial\epsilon_{p}}=\mathbf{v_{p}}\frac{\partial
f^{0}(\epsilon_{p})}{\partial\epsilon_{p}}$ and after several simplifying
steps, the perturbed distribution is [93]

$\displaystyle\delta f$
$\displaystyle=\frac{\mathbf{q}\cdot\mathbf{v_{p}}}{\omega-\mathbf{q}\cdot\mathbf{v_{p}}}\left(-\frac{\partial
f^{0}(\epsilon_{p})}{\partial\epsilon_{p}}\right)[-e\phi_{tot}(\mathbf{q},\omega)],$
(S2)

and the induced charge density is subsequently

$\displaystyle\rho_{ind}(\mathbf{q},\omega)$
$\displaystyle=\frac{g}{\mathcal{V}}\sum_{\mathbf{k}}\frac{\mathbf{q}\cdot\mathbf{v_{p}}}{\omega-\mathbf{q}\cdot\mathbf{v_{p}}}\left(-\frac{\partial
f^{0}(\epsilon_{p})}{\partial\epsilon_{p}}\right)[-e\phi_{tot}(\mathbf{q},\omega)],$
(S3)

where $\mathcal{V}$ is the normalization volume and $g$ is the degeneracy.
From the induced charge density, we can extract the dynamical polarizability
function, also called the density-density response function,
$\Pi_{\rho,\rho}(\mathbf{q},\omega)=\rho_{ind}/(-e\phi_{tot})$ as [94]

$\displaystyle\Pi_{\rho,\rho}(\mathbf{q},\omega)$
$\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p}}\frac{\mathbf{q}\cdot\mathbf{v_{p}}}{\omega-\mathbf{v_{p}\cdot
q}}\frac{\partial f^{0}(\epsilon_{p})}{\partial\epsilon_{p}}.$ (S4)

Note that we are dropping the $\rho,\rho$ subscript for the rest of the text,
using $\Pi_{\rho,\rho}(\mathbf{q},\omega)=\Pi(\mathbf{q},\omega)$. The
integral can then be transformed from 3-dimensional momentum-space to the one
over solid angle, $\Omega$, and energy [95, 39]:

$\displaystyle\frac{1}{\mathcal{V}}\sum_{\mathbf{p}}$
$\displaystyle\rightarrow\int\frac{d\Omega}{4\pi}d\epsilon
N(\epsilon)\rightarrow\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}\int d\epsilon N(\epsilon),$ (S5)

where $N(\epsilon)=\frac{\epsilon^{2}}{2\pi^{2}\hbar^{3}v_{F}^{3}}$ is the
electron density of states for a WSM/DSM. Thus,

$\displaystyle\Pi(\mathbf{q},\omega)$
$\displaystyle=-g\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}\frac{\mathbf{q}\cdot\mathbf{v_{p}}}{\omega-\mathbf{v_{p}\cdot
q}}\int d\epsilon N(\epsilon)\frac{\partial
f^{0}(\epsilon_{p})}{\partial\epsilon_{p}}.$ (S6)

Note that for low temperature, $kT\ll\epsilon_{F}$
$,\partial_{\epsilon}f^{0}\approx-\delta\left(\epsilon(\mathbf{p})-\mu\right)$
[92], leading to:

$\displaystyle\Pi(\mathbf{q},\omega)$
$\displaystyle=gN(\mu)\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}\frac{\mathbf{q}\cdot\mathbf{v_{p}}}{\omega-\mathbf{v_{p}\cdot
q}}.$ (S7)

To solve for the longitudinal polarizability,
$\Pi_{\rho_{z},\rho_{z}}({q},\omega)$, we will consider the case where the
electric field, $\mathbf{E}$, is parallel to the direction of propagation,
$\mathbf{E}||\mathbf{q}||\mathbf{\hat{z}}$. Moving forward we will again drop
the $\rho,\rho$ subscript and use $\Pi({q},\omega)$. Note
$\nabla_{p}\epsilon=\mathbf{v_{p}}=v_{F}(sin{\theta}\cos{\phi}$,
$\sin\theta\sin\phi$, $\cos{\theta})$, thus $v_{z}=v_{F}\cos{\theta}$. We
assume an isotropic electron band structure, specifically an isotropic Fermi
surface such that $v_{F}\hat{x}=v_{F}\hat{y}=v_{F}\hat{z}$. This assumption
has been used in the RPA for WSM/DSM permittivity and the implications of
anisotropic bandstructure are discussed in Ref. [59]. This assumption allows
us to consider $\mathbf{q}=q\hat{z}$ for simplicity of integration [96]

$\displaystyle\Pi({q},\omega)$
$\displaystyle=gN(\mu)\frac{1}{4\pi}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\frac{{q}{v_{F}\cos\theta}}{\omega-{qv_{F}\cos\theta}}.$
(S8)

Let $a=\frac{qv_{F}}{\omega}\cos\theta$ ,
$\frac{da}{d\theta}=-\frac{qv_{F}}{\omega}\sin\theta$,
$d\theta\sin\theta=-da\frac{\omega}{qv_{F}}$. Then,

$\displaystyle\Pi({q},\omega)$
$\displaystyle=gN(\mu)\frac{1}{4\pi}\int_{0}^{2\pi}d\phi\int_{qv_{F}/\omega}^{-qv_{F}/\omega}\left(\frac{-\omega}{qv_{F}}\right)da\frac{a}{1-a},$
(S9)
$\displaystyle=gN(\mu)\frac{1}{4\pi}\left(\frac{\omega}{qv_{F}}\right){2\pi}\left[a+\ln{(1-a)}\right]_{qv_{F}/\omega}^{-qv_{F}/\omega}.$

Assuming a complex-valued logarithm, this solves to

$\displaystyle\Pi({q},\omega)$
$\displaystyle=-gN(\mu)\left(\frac{\omega}{qv_{F}}\frac{1}{2}\ln{\frac{\omega-
qv_{F}}{\omega+qv_{F}}}+1\right).$ (S10)

This is the semiclassical, nonlocal dynamical polarizability result as
presented in the main text, $\Pi^{SC,NL}({q},\omega)$ (Eq. 5). For
thoroughness, we define the real and imaginary parts

$\displaystyle\mbox{Re}~{}\Pi({q},\omega)$
$\displaystyle=\mbox{Re}\left[-g\frac{\mu^{2}}{2\pi^{2}\hbar^{3}v_{F}^{3}}\left(\frac{\omega}{qv_{F}}\frac{1}{2}\ln{\frac{\omega-
qv_{F}}{\omega+qv_{F}}}+1\right)\right]$ (S11)
$\displaystyle\mbox{Im}~{}\Pi({q},\omega)$
$\displaystyle=\mbox{Im}\left[-g\frac{\mu^{2}}{2\pi^{2}\hbar^{3}v_{F}^{3}}\left(\frac{\omega}{qv_{F}}\frac{1}{2}\ln{\frac{\omega-
qv_{F}}{\omega+qv_{F}}}+1\right)\right]$ $\displaystyle=0,~{}~{}\omega>v_{F}q$

In the semiclassical regime, it is assumed that the energy scales of
$\hbar\omega$ and $v_{F}q$ are much less than $\mu$, so the model here is
compared to the RPA result [59] over the $2\mu-\hbar v_{F}q-\hbar\omega>0$
domain in the main text Fig. 2. Qualitatively, the semiclassical model shows
good agreement with the RPA model.

The longitudinal polarizability can then be converted to longitudinal optical
conductivity, in this case $\sigma_{zz}$, via [65]

$\displaystyle{\mbox{Im}\sigma_{zz}}(q,\omega)$
$\displaystyle=\frac{ie^{2}\omega}{q^{2}}\mbox{Re}\Pi(q,\omega),$ (S12)
$\displaystyle=\frac{ie^{2}\omega}{q^{2}}\mbox{Re}\left[-gN(\mu)\left(\frac{\omega}{qv_{F}}\frac{1}{2}\ln{\frac{\omega-
qv_{F}}{\omega+qv_{F}}}+1\right)\right],$
$\displaystyle{\mbox{Re}\sigma_{zz}}(q,\omega)$
$\displaystyle=-\frac{e^{2}\omega}{q^{2}}\mbox{Im}\Pi(q,\omega),$ (S13)
$\displaystyle=\frac{e^{2}\omega}{q^{2}}\mbox{Im}\left[gN(\mu)\left(\frac{\omega}{qv_{F}}\frac{1}{2}\ln{\frac{\omega-
qv_{F}}{\omega+qv_{F}}}+1\right)\right].$

As shown in Supplementary Fig. S1, the resulting optical conductivity in the
semiclassical model aligns with the RPA model best in the region $2\mu-\hbar
v_{F}q-\hbar\omega>0$ and $\omega<v_{F}q$ . As expected, the semiclassical
model has decreased alignment with the RPA result at higher energy,
$\omega>v_{F}q$.

Figure S1: The imaginary part of the optical conductivity, $\mbox{Im}\sigma$,
for the RPA calculation (a) and for the semiclassical nonlocal (SC,NL)
derivation in this work (b). In (b), the semiclassical $\sigma^{SC,NL}$ is
plotted on the $2\mu-\hbar v_{F}q-\hbar\omega>0$ domain, corresponding to the
area below the white dashed line, and the RPA solution is kept for comparison
for $2\mu-\hbar v_{F}q-\hbar\omega<0$ . In (c), the difference between the RPA
and semiclassical $\mbox{Im}\sigma$ is plotted. All plots are in units of
$e^{2}gk_{F}/4\pi^{2}\hbar$.

Finally, recall that the semiclassical $\Pi(q,\omega)$ derived here accounts
for only intra-band transitions. The material permittivity,
$\varepsilon(q,\omega)$, accounting for just intra-band transitions is found
via [61, 97, 58, 59, 15]

$\displaystyle\varepsilon(q,\omega)$
$\displaystyle=\varepsilon_{\infty}(1-V_{q}\Pi^{intra}(q,\omega)),$ (S14)
$\displaystyle=\varepsilon_{\infty}+i\frac{\sigma^{intra}}{\varepsilon_{0}\omega},$

where $\varepsilon_{\infty}$ is the background permittivity and
$V_{q}=e^{2}/(\varepsilon_{\infty}\varepsilon_{0}q^{2})$ is the Fourier
transform of the Coulomb potential energy in SI units [97]. The resulting
local permittivity values ($q\rightarrow 0$) of the semiclassical and RPA
approaches show good agreement (Supplementary Fig. S2(a)): in the usual Drude
fashion, the semiclassical Drude result lacks the peak at
$\hbar\omega/\epsilon_{F}=2$, which corresponds to inter-band absorption [61].

Figure S2: (a) The real part of the WSM/DSM permittivity in the local
approximation, $\varepsilon(\omega)$. All plots are for the case of
$\epsilon_{F}=0.15$ eV, $v_{F}=1\times 10^{6}$ m/s, $g=4$,
$\varepsilon_{\infty}=13$, and $\varepsilon_{b}=12$ characteristic to Cd3As2
[61]. In the RPA $\epsilon_{max}=3\epsilon_{F}$.

Note that for combined inter- and intra-band contributions, the relationship
between $\Pi(q,\omega)$ and $\varepsilon(q,\omega)$ is

$\displaystyle\varepsilon(q,\omega)$
$\displaystyle=\varepsilon_{b}(1-V_{q}(\Pi^{intra}(q,\omega)+\Pi^{inter}(q,\omega))),$
(S15)
$\displaystyle=\varepsilon_{b}+i\frac{(\sigma^{intra}(q,\omega)+\sigma^{inter}(q,\omega))}{\varepsilon_{0}\omega},$

where $\varepsilon_{b}$ is a function of $\varepsilon_{\infty}$ [61].

Finally, the effects of loss via scattering are accounted for in the
polarizability by introducing a phenomenological relaxation time parameter
$\tau$, which accounts for losses due to electron scattering on impurities and
lattice defects as well as electron-phonon scattering [98, 99]:

$\displaystyle\Pi_{\tau}({q},\omega)$
$\displaystyle=\frac{(1+i/\omega\tau)\Pi(q,\omega+i/\tau)}{1+(i/\omega\tau)\Pi(q,\omega+i/\tau)/\Pi(q,0)}.$
(S16)

Thus,

$\displaystyle\Pi_{\tau}({q},\omega)$
$\displaystyle=\frac{(1+i/\omega\tau)gN(\mu)\left(-\frac{\omega+i/\tau}{qv_{F}}\frac{1}{2}\ln{\frac{\omega+i/\tau+qv_{F}}{\omega+i/\tau-
qv_{F}}}+1\right)}{1+(i/\omega\tau)\left(-\frac{\omega+i/\tau}{qv_{F}}\frac{1}{2}\ln{\frac{\omega+i/\tau+qv_{F}}{\omega+i/\tau-
qv_{F}}}+1\right)}.$ (S17)

###  I.2 Local model

For completeness, we show that this same approach leads to the well-known
local Drude approximation for WSM/DSMs when $\omega\gg qv$ is assumed. The
nonlocal semiclassical model can be reduced to the local one by using the
Taylor expansion of the polarizability function in powers of $q$ around
$q\rightarrow 0$ [65]. Starting from Eq. S9, we obtain:

$\displaystyle\frac{a}{1-a}\approx a+a^{2}+O[a]^{3}.$ (S18)

Inserting this term into the expression for the polarizability, we arrive at
the local semiclassical model:

$\displaystyle\Pi({q},\omega)$ $\displaystyle\approx
gN(\mu)\frac{1}{4\pi}\left(\frac{\omega}{qv_{F}}\right){2\pi}\left[\frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right]_{-qv_{F}/\omega}^{qv_{F}/\omega},$
(S19)
$\displaystyle=\frac{1}{3}gN(\mu)\frac{q^{2}v_{F}^{2}}{\omega^{2}}=\frac{1}{6}\frac{{q}^{2}g}{\omega^{2}}\frac{\mu^{2}}{\pi^{2}\hbar^{3}v_{F}}.$

The longitudinal polarizability is converted to longitudinal optical
conductivity:

$\displaystyle\mbox{Im}~{}{\sigma_{zz}}(q,\omega)$
$\displaystyle=i\frac{1}{6}\frac{ge^{2}}{\omega}\frac{v_{F}k_{F}^{2}}{\pi^{2}\hbar}.$
(S20)

This is equivalent to the known local optical conductivity for WSMs and DSMs
$\sigma_{zz}=\frac{i\varepsilon_{\infty}}{4\pi\omega}\omega_{p}^{2}$, where
$\omega_{p}=\frac{2k_{F}^{2}v_{F}e^{2}g}{3\hbar\pi\varepsilon_{\infty}}$ [61].
Accounting for loss, these expressions take the following form:

$\displaystyle\Pi_{\tau}({q},\omega)$
$\displaystyle=\frac{1}{6}\frac{{q}^{2}g}{\omega(\omega+i/\tau)}\frac{\mu^{2}}{\pi^{2}\hbar^{3}v_{F}},$
(S21)

$\displaystyle\sigma_{zz,\tau}$
$\displaystyle=\frac{i\varepsilon_{\infty}}{4\pi(\omega+i/\tau)}\omega_{p}^{2}.$
(S22)

##  II Supplemental Note 2: Longitudinal polarizability and optical
conductivity in DSM and WSM with current bias

###  II.1 Nonlocal case

Here, we derive the dynamical longitudinal polarizability,
$\Pi_{\rho,\rho}^{u}(q,\omega)$, and optical conductivity,
$\sigma^{u}(q,\omega)$, of a bulk Dirac semimetal and Weyl semimetal in a
drift current-biased state. As in Supplementary Note I, the quasiparticle
distribution function is modeled as a combination of the equilibrium
distribution, $f^{u}(\mathbf{p})$, and a perturbation to the distribution,
$\delta f(\mathbf{p})$, $f(\mathbf{p})=f^{u}(\mathbf{p})+\delta
f(\mathbf{p})$, assuming an externally applied electric field,
$\mathbf{E}=\mathbf{E_{0}}e^{-i\omega t+i\mathbf{q}\cdot\mathbf{r}}$. In the
case of an applied DC current, the equilibrium distribution is a skewed Fermi
distribution of the form [39, 43]:

$f^{u}(\mathbf{p})=\frac{1}{e^{(E(\mathbf{p})-\mathbf{u}\cdot\mathbf{p}-\mu^{u})/k_{B}T}+1},$
(S23)

where $E(\mathbf{p})=v|\mathbf{p}|$. We will express
$\nabla_{p}f^{u}(\mathbf{p})$ as an integral over the independent energy
variable, $\epsilon$ [39]

$\displaystyle\nabla_{p}f^{u}(\mathbf{p})=\int
d\epsilon\delta(\epsilon-E(\mathbf{p}))(\mathbf{v_{p}-u})\frac{\partial
f^{u}(\epsilon,\mathbf{p})}{\partial\epsilon},$ (S24)

where

$f^{u}(\mathbf{p})=\frac{1}{e^{(\epsilon-\mathbf{u}\cdot\mathbf{p}-\mu^{u})/k_{B}T}+1},$
(S25)

and note that $v_{\mathbf{p}}=\frac{v^{2}\mathbf{p}}{\epsilon}$. From the
above equations, we can express the polarizability as

$\displaystyle\Pi^{u}(\mathbf{q},\omega)$
$\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p}}\frac{\mathbf{q}\cdot\nabla_{p}f^{u}(\mathbf{p})}{\omega-\mathbf{v_{p}\cdot
q}},$ (S26) $\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p}}\int
d\epsilon\delta(\epsilon-E(\mathbf{p}))\frac{v^{2}\mathbf{q\cdot
p}-\epsilon\mathbf{q\cdot
u}}{\epsilon\omega-v^{2}\mathbf{p}\cdot\mathbf{q}}\frac{\partial
f^{u}(\epsilon,\mathbf{p})}{\partial\epsilon}.$

To solve Eq. S26, we will perform a quasi-Lorentz transformation (QLT) with
the below transformations for current traveling in the z-direction (this
transformation is justified in the main text):

$\epsilon_{0}(p)=\gamma\left(\epsilon(p)-up^{z}\
\right),~{}p_{0}^{z}=\gamma\left(p^{z}-\frac{u}{v_{F}^{2}}\epsilon(p)\right),~{}p_{0}^{x}=p^{x},~{}p_{0}^{y}=p^{y},$
(S27)

$\displaystyle\omega_{0}=\gamma(\omega-uq_{z}),~{}q_{0}^{z}$
$\displaystyle=\gamma(q^{z}-\frac{u}{v_{F}^{2}}\omega),~{}q_{0}^{x}=q^{x},~{}q_{0}^{y}=q^{y},~{}$
(S28)

$\gamma=(1-u^{2}/v_{F}^{2})^{-1/2}.$ (S29)

Where $\gamma$ is the QLT factor and the subscript “$0$” represents
measurements in the moving frame with velocity $u$ and the subscript “$u$”
represents measurements in the stationary frame (following the notation of
[43]).

The perturbed density is transformed as

$f^{u}(\epsilon)=\frac{1}{e^{(\epsilon-\mathbf{u}\cdot\mathbf{p}-\mu^{u})/k_{B}T}+1}=\frac{1}{e^{(\epsilon_{0}-\mu_{0})/\gamma
k_{B}T}+1},$ (S30)

where we have introduced $\mu_{0}=\gamma\mu^{u}$. The chemical potential must
be transformed such that the quasiparticle density, $n$, is conserved:
$n_{0}=n/\gamma$ [43]. The quasiparticle density of a WSM/DSM is
$n=\int_{0}^{\epsilon_{F}}N(\epsilon)d\epsilon$, where the electron density of
states is $N(\epsilon)=\epsilon^{2}/({2\pi^{2}\hbar^{3}v_{F}^{3}})$. At 0K we
can show that the quasiparticle density is
$n=\epsilon_{F}^{3}/(6\pi^{2}\hbar^{3}v_{F}^{3})$, and thus the carrier
density and chemical potential follow the relationship $n\propto\mu^{3}$
[100]. Therefore, to conserve carrier density the chemical potential must
transform as $\mu_{0}=\mu/\gamma^{1/3}$ and subsequently
$\mu^{u}=\mu_{0}/\gamma=\mu/\gamma^{4/3}$.

Next, note that according to our definition [39]

$\frac{\partial f^{u}(\epsilon)}{\partial\epsilon}=\gamma\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S31)

Additionally, the transformation of $\delta(\epsilon-v|p|)$ is found by making
use of the delta function identity from quantum field theory [101], defined
using an arbitrary function $g(x)$, the identity is

$\delta(g(x)-g(x_{0}))=\frac{1}{|g^{\prime}(x_{0})|}\delta(x-x_{0}).$ (S32)

We express the QLT of the energy, $\epsilon_{0}$, as

$\displaystyle g(x)$ $\displaystyle=\gamma(x-up(x)),$ (S33) $\displaystyle
g^{\prime}(x)$ $\displaystyle=\gamma(1-u\frac{dp(x)}{dx}),$

where $g(x)=\epsilon_{0}$ and $x=\epsilon$, and via $\epsilon=v|p|$ we know
that $p(x)=\epsilon/v=x/v$. Using the delta function identity, we find

$\displaystyle\delta(\epsilon_{0}-v|p_{0}|)$
$\displaystyle=\frac{1}{|\gamma(1-u/v)|}\delta(\epsilon-v|p|),$ (S34)
$\displaystyle=\frac{\epsilon_{0}}{\epsilon}\delta(\epsilon-v|p|).$

Thus,

$\displaystyle\Pi^{u}(\mathbf{q},\omega)$
$\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p}}\int
d\epsilon\frac{\epsilon}{\epsilon_{0}}\delta(\epsilon_{0}-v|p_{0}|)\frac{v^{2}\mathbf{q\cdot
p}-\epsilon\mathbf{q\cdot
u}}{\epsilon\omega-v^{2}\mathbf{p}\cdot\mathbf{q}}\gamma\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S35)

Under the QLT $\epsilon\omega-v_{F}^{2}\mathbf{p\cdot
q}=\epsilon_{0}\omega_{0}-v_{F}^{2}\mathbf{p_{0}\cdot q_{0}}$ and
$v^{2}p^{z}-\epsilon u=v^{2}p_{0}^{z}/\gamma$ [39]. To extract the
longitudinal conductivity,
$\Pi^{u}_{\rho_{z},\rho_{z}}({q},\omega)=\Pi^{u}({q},\omega)$, we again
consider the longitudinal case of $\mathbf{E}||\mathbf{q}||\mathbf{\hat{z}}$:

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p_{0}}}\int
d\epsilon_{0}\frac{\epsilon}{\epsilon_{0}}\delta(\epsilon_{0}-v|p_{0}|)\frac{qv_{F}^{2}p_{0}^{z}/\gamma}{\epsilon_{0}\omega_{0}-p_{0}^{z}v_{F}^{2}q_{0}}\gamma\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S36)

Solving the $\epsilon_{0}$ integral over the delta function we get
$\epsilon_{0}=v|\mathbf{p_{0}}|$, leading to

$\displaystyle\Pi^{u}({q},\omega)$ $\displaystyle=$
$\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p_{0}}}\frac{\epsilon}{\epsilon_{0}}\frac{qv_{F}^{2}p_{0}^{z}}{\epsilon_{0}\omega_{0}-p_{0}^{z}v_{F}^{2}q_{0}}\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S37)

Note that $\frac{\epsilon}{\epsilon_{0}}=\gamma(1+up_{0}^{z}/\epsilon_{0})$,
and thus

$\displaystyle\Pi^{u}({q},\omega)$ $\displaystyle=$
$\displaystyle=-\frac{g}{\mathcal{V}}\sum_{\mathbf{p_{0}}}\gamma(1+\frac{up_{0}^{z}}{\epsilon_{0}})\frac{qv_{F}^{2}p_{0}^{z}}{\epsilon_{0}\omega_{0}-p_{0}^{z}v_{F}^{2}q_{0}}\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S38)

The integral can then be transformed from 3D $p_{0}$-space to over a constant
energy surface $\epsilon_{0}$ via Eq. S5 [95, 39]

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=-g\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}\int
d\epsilon_{0}N(\epsilon_{0})~{}\gamma\left(1+\frac{up_{0}\cos{\theta}}{\epsilon_{0}}\right)\frac{qv_{F}^{2}p_{0}\cos{\theta}}{\epsilon_{0}\omega_{0}-p_{0}\cos{\theta}v_{F}^{2}q_{0}}\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S39)

This can be evaluated for low temperature $kT\ll\mu$ using ${\partial
f^{0}(\epsilon_{0})}/{\partial\epsilon_{0}}\approx-\delta\left(\epsilon_{0}-\mu_{0}\right)$
[101, 39]

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=gN(\mu_{0})\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}~{}\gamma\left(1+\frac{up_{0}\cos{\theta}}{\mu_{0}}\right)\frac{qv_{F}^{2}p_{0}\cos{\theta}}{\mu_{0}\omega_{0}-p_{0}\cos{\theta}v_{F}^{2}q_{0}}.$
(S40)

Note that $\frac{1}{v_{F}}=\frac{p_{0}(\mu_{0})}{\mu_{0}}$, and thus

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=gN(\mu_{0})2\pi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}~{}\gamma\left(1+\frac{u\cos{\theta}}{v_{F}}\right)\frac{v_{F}q\cos{\theta}}{\omega_{0}-v_{F}q_{0}\cos{\theta}}.$
(S41)

Let $a=\cos\theta$ , $\frac{da}{d\theta}=-\sin\theta$, and
$d\theta\sin\theta=-da$:

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=gN(\mu_{0})\frac{\gamma}{2}\int_{1}^{-1}-da~{}\left(1+\frac{ua}{v_{F}}\right)\frac{v_{F}qa}{\omega_{0}-v_{F}q_{0}a},$
(S42)
$\displaystyle=-gN(\mu_{0})\frac{1}{2}\frac{q\gamma(q_{0}v_{F}^{2}+u\omega_{0})}{q_{0}^{3}v_{F}^{3}}\left(\omega_{0}\ln\left(\frac{\omega_{0}-q_{0}v_{F}}{\omega_{0}+q_{0}v_{F}}\right)+2q_{0}v_{F}\right),$
$\displaystyle=-gN(\mu_{0})\frac{q\gamma(q_{0}+\frac{u}{v_{F}^{2}}\omega_{0})}{q_{0}^{2}}\left(\frac{\omega_{0}}{2q_{0}v_{F}}\ln\left(\frac{\omega_{0}-q_{0}v_{F}}{\omega_{0}+q_{0}v_{F}}\right)+1\right),$
$\displaystyle=-gN(\mu_{0})\frac{q^{2}}{q_{0}^{2}}\left(\frac{\omega_{0}}{2q_{0}v_{F}}\ln\left(\frac{\omega_{0}-q_{0}v_{F}}{\omega_{0}+q_{0}v_{F}}\right)+1\right).$

The longitudinal polarizability is converted to longitudinal optical
conductivity as follows:

$\displaystyle{\mbox{Im}\sigma^{u}_{zz}}(q,\omega)$
$\displaystyle=\frac{ie^{2}\omega}{q^{2}}\mbox{Re}\Pi^{u}(q,\omega),$ (S43)
$\displaystyle=-\frac{ie^{2}\omega}{q_{0}^{2}}gN(\mu_{0})\left(\frac{\omega_{0}}{2q_{0}v_{F}}\ln\left(\frac{\omega_{0}-q_{0}v_{F}}{\omega_{0}+q_{0}v_{F}}\right)+1\right).$

Comparing Eq. S43 to Eq. S12, we have shown that the longitudinal optical
conductivity is transformed under Fizeau drag as:

$\displaystyle\sigma_{zz}^{u}(\omega,q,\mu)$
$\displaystyle=\frac{\omega}{\omega_{0}}\sigma_{zz}(\omega_{0},q_{0},\mu_{0}),$
(S44)

where

$\displaystyle\omega_{0}=\gamma(\omega-uq_{z}),~{}q_{z,0}$
$\displaystyle=\gamma(q_{z}-\frac{u}{v_{F}^{2}}\omega),~{}q_{x,0}=q_{x},~{}q_{y,0}=q_{y},~{}\mu_{0}=\frac{\mu}{\gamma^{1/3}}.$
(S45)

###  II.2 Local case

For thoroughness and comparison, we show that the same Fizeau drag derivation
can be performed for the local case, making use of Taylor series expansion
around $q_{0}\rightarrow 0$ starting from equation S38:

$\displaystyle\frac{qv_{F}^{2}p_{0}^{z}}{\epsilon_{0}\omega_{0}-p_{0}^{z}v_{F}^{2}q_{0}}=\frac{qv_{F}^{2}p_{0}^{z}}{\epsilon_{0}\omega_{0}}+\frac{qq_{0}v_{F}^{4}(p_{0}^{z})^{2}}{\epsilon_{0}^{2}\omega_{0}^{2}}+O[q]^{2}$
(S46)

$\displaystyle(1+\frac{up_{0}^{z}}{\epsilon_{0}})\frac{qv_{F}^{2}p_{0}^{z}}{\epsilon_{0}\omega_{0}-p^{x}_{0}v_{F}^{2}q_{0}}$
$\displaystyle\approx(1+\frac{up_{0}^{z}}{\epsilon_{0}})\left(\frac{qv_{F}^{2}p_{0}^{z}}{\epsilon_{0}\omega_{0}}+\frac{qq_{0}v_{F}^{4}(p_{0}^{z})^{2}}{\epsilon_{0}^{2}\omega_{0}^{2}}\right)$
(S47)
$\displaystyle\approx\frac{quv_{F}^{2}(p_{0}^{z})^{2}}{\epsilon^{2}_{0}\omega_{0}}+\frac{qq_{0}v_{F}^{4}(p_{0}^{z})^{2}}{\epsilon_{0}^{2}\omega_{0}^{2}}$
$\displaystyle=\frac{q^{2}v_{F}^{4}(p_{0}^{z})^{2}}{\epsilon^{2}_{0}\omega^{2}_{0}}\frac{1}{\gamma}.$

Thus,

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle\approx-\frac{g}{\mathcal{V}}\sum_{\mathbf{p_{0}}}~{}\frac{q^{2}v_{F}^{4}(p_{0}^{z})^{2}}{\epsilon^{2}_{0}\omega^{2}_{0}}\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S48)

The integral can then be transformed from 3-dimensional $p$-space to over a
constant energy surface via Eq. S5

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=-g\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta
sin\theta}{4\pi}\int d\epsilon
N(\epsilon)~{}\frac{q^{2}v_{F}^{4}(p_{0}(\epsilon_{0})\cos{\theta})^{2}}{\epsilon^{2}_{0}\omega^{2}_{0}}\frac{\partial
f^{0}(\epsilon_{0})}{\partial\epsilon_{0}}.$ (S49)

As for the nonlocal case, this is evaluated for low-temperature $kT\ll\mu$
using ${\partial
f^{0}(\epsilon_{0})}/{\partial\epsilon_{0}}\approx-\delta\left(\epsilon_{0}-\mu_{0}\right)$

$\displaystyle\Pi^{u}({q},\omega)$
$\displaystyle=g\frac{q^{2}v_{F}^{2}}{\omega^{2}_{0}}N(\mu_{0})\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\frac{d\theta}{4\pi}\sin{\theta}\cos^{2}{\theta},$
(S50)
$\displaystyle=\frac{1}{3}N(\mu_{0})g\frac{q^{2}v_{F}^{2}}{\omega^{2}_{0}},$
$\displaystyle=\frac{1}{3}\frac{g\mu_{0}^{2}}{2\pi^{2}\hbar^{3}v_{F}}~{}\frac{q^{2}}{\omega^{2}_{0}}.$

The longitudinal polarizability is converted to longitudinal optical
conductivity

$\displaystyle\sigma^{u}_{zz}(q,\omega)$
$\displaystyle=\frac{\omega}{\omega_{0}}\frac{ie^{2}g\mu_{0}^{2}}{6\pi^{2}\hbar^{3}\omega_{0}v_{F}},$
(S51) $\displaystyle=\frac{\omega}{\gamma(\omega-
uq)}\frac{ie^{2}g\mu_{0}^{2}}{6\pi^{2}\hbar^{3}v_{F}\cdot\gamma(\omega-uq)}.$

When compared to the non-current carrying case, it is clear that the
longitudinal optical conductivity is transformed again following equations S44
and S45.

In Suppl. Fig. S3(a-d) we plot the SPP dispersion for lossless Cd3As2 for
increasing current biases using the local Drude, Eq. 4, nonlocal Drude, Eq. 5,
and RPA, Eq. 1, dielectric models. The RPA and nonlocal Drude models closely
match, while the local Drude model does not align with either result. With
increased carrier velocity, the local and nonlocal optical models increasingly
diverge, resulting in vastly different plasmon frequencies, where the local
model overestimates the magnitude of Fizeau shift. This emphasizes the
importance of accounting for nonlocality in Fizeau drag calculations.

The nonlocal predictions for Fizeau drag differs notably from the prediction
of truly unidirectional SPP transport which can be made when invoking the non-
realistic local Drude mode. The local Drude model predicts that the counter-
propagating mode will never bend upward to compete with the co-propagating
mode, facilitating unidirectionality.

Figure S3: The SPP dispersion for Cd3As2 ($v_{F}=1\times 10^{6}$,
$\epsilon_{F}=0.15$ eV, and $g=4$) for the local Drude [61], Eq. 4, nonlocal
Drude, Eq. 5, and the RPA dielectric model presented in Ref. [58, 59], Eq. 1,
for the (a) unbiased case and (b) $u=0.1v_{F}$, (c) $u=0.3v_{F}$, and (d)
$u=0.5v_{F}$.

##  III Supplemental Note 3: Complex-q solving method

All the SPP mode solutions shown in this manuscript have been found by
searching over a complex $q=\mbox{Re}(q)+\mbox{Im}(q)$ space as shown in
Suppl. Fig. S4. The solutions corresponding to lossy non-propagating modes,
with $\mbox{Re}(q)/\mbox{Im}(q)<1$, are included in the main figures, while
the solutions representing gain, $\mbox{Re}(q)/\mbox{Im}(q)>1$, are assumed to
be unphysical [44], but are included in Supplementary Fig. S4 in black. The
SPP dispersion for Cd3As2 without any Fizeau drag is shown in Supplementary
Fig. S5.

Figure S4: Complex q-space solution for increasing loss. The equifrequency
complex wavevector solution space for $\omega=0.337\epsilon_{F}$ for (a)
$\tau=2.1\times 10^{-10}s$, (b) $\tau=2.1\times 10^{-11}$s, (c)
$\tau=2.1\times 10^{-12}$s and the corresponding SPP dispersion for each in
(d), (e), and (f) respectively. The solid line represented the $\mbox{Re}(q)$
solution and the dashed represented the $\mbox{Im}(q)$ solution. The black
corresponds to the solution representing gain,
$\mbox{Re}(q)/\mbox{Im}(q)>1$.The same Cd3As2 parameters are used as in Fig.
4.

In Suppl. Fig. S5(a-c) we show the SPP dispersion for Cd3As2 for decreasing
scattering period from $\tau=2.1\times 10^{-10}$s to $2.1\times 10^{-12}$s.

Figure S5: The SPP dispersion for Cd3As2 ($v_{F}=1\times 10^{6}$,
$\epsilon_{F}=0.15$ eV, and $g=4$) for the nonlocal Drude model derived in Eq.
5 under no DC-bias in the (a) $\tau=2.1\times 10^{-10}$, (b) $\tau=2.1\times
10^{-11}$ and (c) $\tau=2.1\times 10^{-12}$ case.

##  IV Supplementary Note 4: Comparison to graphene Fizeau drag

Here we plot the SPP dispersion for Fizeau drag in graphene using the
derivations presented in Ref. [43] for the air, Suppl. Fig S6(a), and hBN
encapsulated, Suppl. Fig S6(b), cases. These are shown alongside the SPP
dispersion for the 3D DSM Cd3As2. In all the panels, the dispersion
characteristics have been calculated by using nonlocal optical models in the
lossless case.

Figure S6: The SPP dispersion for Fizeau drag in (a) Graphene in air (b)
Graphene encapsulated in hBN and (c) the 3D DSM Cd3As2. The graphene is
modeled following the equations in Ref. [43], using $n=2.88\times 10^{12}$,
$v_{F}=1\times 10^{6}$ms-1, $E_{F}=198$ meV. The hBN is modeled using the
parameters from Ref. [102]. The Cd3As2 is modeled with the same parameters as
in Fig. 3.