]Department of Physics, Washington University in St. Louis, MO 63104, USA
]Department of Physics, Washington University in St. Louis, MO 63104, USA
]Department of Physics, Washington University in St. Louis, MO 63104, USA ]The
Institute for Functional Intelligent Materials (I-FIM), National University of
Singapore, 4 Science Drive 2, Singapore 117544 ]Department of Physics and
Astronomy, University of Missouri, Columbia, Missouri 65211, USA

# Observability of cyclotron resonance in the hydrodynamic regime of bilayer
graphene

Joseph R. Cruise  [  Alexander Seidel  [  Erik Henriksen  [  Giovanni Vignale
[  [

###### Abstract

We offer theoretical predictions for the frequency of the resonant frequency
of transport for the hydrodynamic description of bilayer graphene, as well as
provide quantification for the relative strength of this signal throughout
phase space. Our calculations are based on classical fluid dynamics equations
derived from the Boltzmann equation for bilayer graphene in [1], and suggest
that while this resonance is accessible to current experimental techniques,
the same mechanism which causes the hydrodynamic resonance to differ from the
Fermi liquid value is responsible for a significant broadening of the peak.

††preprint: APS/123-QED

##  I  Introduction

Since its isolation in 2004 [2], graphene in its various manifestations has
demonstrated a remarkable potential for hosting rich phases of quantum matter
[3, 4, 5, 6, 7]. Much of this novel behavior is attributable to the Dirac
points in the Brillouin zone, where the excitations behave like (2+1)-d
relativistic quasiparticles [8, 9]. It has been suggested theoretically and
demonstrated experimentally that these massless Dirac fermions are
realistically described by electron hydrodynamics (EH)[10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23] . The recent experimental identification of
this novel phase in monolayer graphene has generated interest in other related
systems–such as bilayer graphene–for evidence of hydrodynamics [1, 24, 25],
although this regime has been less thoroughly explored.

The idea behind EH is tantalizingly simple: charged particles, interacting
only amongst themselves through the conservative Coulomb force, will enter a
thermal equilibrium, and so on sufficiently long time scales will be described
by classical fluid dynamics equations. (See [26, 16] for an overview of
hydrodynamics in graphene). It is worth emphasizing that this purely classical
description is a consequence only of statistical mechanics, and so holds
whether the constituent particles are quantum mechanical or not. Hydrodynamics
therefore yields analytically tractable classical results even for highly
correlated quantum mechanical systems. Unfortunately, the generality of
hydrodynamics belies how stringent the requirement of a Coulomb-dominated
phase is.

Graphene distinguishes itself among other candidate materials for
hydrodynamics by virtue of its stiffness, which reduces electron-phonon
scattering, its small Fermi temperature, which mitigates the issue of Pauli
blocking, and its compatibility with hexagonal boron nitride, which diminishes
the effects of disorder [27, 28]: these characterizations are valid for mono-,
as well as bi- and multilayer graphene, all of which are believed to host
hydrodynamic phases. (Recent many-body calculations suggest hydrodynamics of
AB-stacked bilayer graphene may be even more robust than those of monolayer
graphene [25].) The rarity of all these properties occurring simultaneously in
a single material is clear from the history of EH, which saw a span of over 40
years between the development of the theory and the first experimental
confirmation [29]. Consequentially, the fact that different stackings of
graphene are all capable of hosting EH demonstrates the importance of this
material as a resource for studying this regime.

In practice, the clearest signatures of hydrodynamics pertain to geometry-
dependent viscous effects, such as vortex-induced negative resistances [30],
parabolic Poiseuille flow [31, 32], and ratchet effects in spatially modulated
samples [33]. However, hydrodynamics also features unusual microscopic effects
which do not depend on boundary conditions, such as a cyclotron resonance
which differs from the Fermi liquid prediction of $\omega_{c}=eB/m^{*}$; for
the relativistic hydrodynamics of monolayer graphene, it has been shown that
$\omega_{c}\sim 1/T$ [34]. This non-trivial temperature dependence is made
even more striking in recalling Kohn’s theorem, which implies that the free-
electron value of $eB/m^{*}$ holds across all single-band, massive, Galilean-
invariant quasiparticle excitations [35].

In this paper, we will explore the consequences of hydrodynamics on the
cyclotron resonance in AB stacked bilayer graphene. In particular, we will
compute the longitudinal conductivity $\sigma_{xx}$ in order to search for a
region of phase space where the resonance is substantially removed from the
typical value of $eB/m^{*}$. Our starting point is the hydrodynamic equations
of motion given by Ngyuen et. al. ins [36]. In section II we compute the
resonance analytically, and discuss the regimes where our calculation is
valid; in section III we plot the resulting thermoelectric transport
coefficients, giving clear experimental predictions; we conclude in section
IV.

##  II Analytic results

Following [36], we use the quantum Boltzmann equation (QBE) formalism to
derive fluid dynamics equations for both the electrons and holes in bilayer
graphene. These equations read

$\displaystyle-i\omega\vec{u}^{e}$
$\displaystyle=-\frac{n^{h}}{n}\frac{1}{\tau_{0}}\left({\vec{u}^{e}-\vec{u}^{h}}\right)-\frac{\vec{u}^{e}}{\tau_{d}}+\frac{e}{m_{e}^{*}}\left({\vec{E}+\vec{u}^{e}\times\vec{B}}\right)-\frac{\Lambda^{e}}{m^{*}}k_{B}\nabla
T$ (1) $\displaystyle-i\omega\vec{u}^{h}$
$\displaystyle=\frac{n^{e}}{n}\frac{1}{\tau_{0}}\left({\vec{u}^{e}-\vec{u}^{h}}\right)-\frac{\vec{u}^{h}}{\tau_{d}}-\frac{e}{m^{*}}\left({\vec{E}+\vec{u}^{h}\times\vec{B}}\right)-\frac{\Lambda^{h}}{m_{h}^{*}}k_{B}\nabla
T.$ (2)

Here $\vec{u}^{(e/h)}$, $n^{(e/h)}$, and $\Lambda^{(e/h)}$ are the fluid
velocity, number density, and per-particle entropy of the electron/hole fluid
respectively, and $n=n^{e}+n^{h}$; $\tau_{d}$ is the disorder scattering time
(which accounts for phonon, impurity, and finite size interactions);
$\tau_{0}$ is the Coulomb scattering time; and $m^{*}$ is the effective mass
of electrons/holes 111at low energy and magnetic fields, bilayer graphene is
particle-hole symmetric, and so for simplicity we will assume both electrons
and holes have the same effective mass. As equations (1) and (2) are derived
in the linear response regime, they can be interpreted as a constitutive
relation of the form

$\begin{pmatrix}\vec{J}\\\
\vec{Q}\end{pmatrix}=\begin{pmatrix}\sigma&\Theta\\\ T\Theta&K\\\
\end{pmatrix}\begin{pmatrix}e\vec{E}\\\ k_{B}\vec{\nabla}T\end{pmatrix},$ (3)

for $\vec{J}=e(n^{e}\vec{u}^{e}-n^{h}\vec{u^{h}})$ and
$Q=k_{B}T(\Lambda^{e}n^{e}\vec{u}^{e}+\Lambda^{h}n^{h}\vec{u}^{h})-(\mu/e)\vec{J}$
222Heat currents are usually measured under the condition $\vec{J}=0$, in
which case $\vec{Q}=-\kappa\vec{\nabla}T$ where
$\kappa:=K-T\Theta\sigma^{-1}\Theta$ is the true quantity being measured. It
follows that the longitudinal thermo-electric coefficients are given by

$\displaystyle\sigma_{xx}\left({\tilde{\omega}}\right)$
$\displaystyle=\sigma_{D}\frac{\tilde{\omega}\left({\left({\frac{\delta
n}{n}}\right)^{2}\left({1+\tilde{\omega}}\right)+q(\tilde{\omega})}\right)}{\left({\frac{\delta
n}{n}\omega_{c}\tau_{0}}\right)^{2}+q(\tilde{\omega})^{2}}$ (4)
$\displaystyle\Theta_{xx}(\tilde{\omega})$
$\displaystyle=-\sigma_{D}\frac{\tilde{\omega}\left({\delta\Lambda\left({\left({\frac{\delta
n}{n}}\right)^{2}(1+\tilde{\omega})+q(\tilde{\omega})}\right)+\Lambda\frac{\delta
n}{n}\left({\left({\tilde{\omega}+1}\right)^{2}+\left({\omega_{c}\tau_{0}}\right)^{2}}\right)}\right)}{\left({\frac{\delta
n}{n}\omega_{c}\tau_{0}}\right)^{2}+q(\tilde{\omega})^{2}}$ (5) $\displaystyle
K_{xx}(\tilde{\omega})$
$\displaystyle=-\kappa_{D}\left[\frac{\tilde{\omega}\left({\left({\delta\Lambda^{2}-\delta\Lambda\beta\mu}\right)\left({\left({\frac{\delta
n}{n}}\right)^{2}(1+\tilde{\omega})+q(\tilde{\omega})}\right)+\Lambda\frac{\delta
n}{n}\left({2\delta\Lambda-\Lambda\beta\mu}\right)\left({(\tilde{\omega}+1)^{2}+(\tau_{0}\omega_{c})^{2}}\right)}\right)}{\left({\frac{\delta
n}{n}\omega_{c}\tau_{0}}\right)^{2}+q(\tilde{\omega})^{2}}\right.$ (6)
$\displaystyle\ \
+\left.\frac{\Lambda^{2}\left({(1+\tilde{\omega})q(\tilde{\omega})-\left({\frac{\delta
n}{n}\tau_{0}\omega_{c}}\right)^{2}}\right)}{\left({\frac{\delta
n}{n}\omega_{c}\tau_{0}}\right)^{2}+q(\tilde{\omega})^{2}}\right]$

for

$q(\omega):=\omega^{2}+\omega+(\tau_{0}\omega_{c})^{2}.$ (7)

Here, we employ the notation: $\sigma_{D}=ne\tau_{0}/m$ and
$\kappa_{D}=nk_{B}^{2}T\tau_{0}/m$ are the Drude model electric and thermal
conductivities respectively; $\delta n=n^{e}-n^{h}$,
$\Lambda=(\Lambda^{e}+\Lambda^{h})/2$ and
$\delta\Lambda=(\Lambda^{e}-\Lambda^{h})/2$;
$\tilde{\omega}=-i\omega\tau_{0}+\tau_{0}/\tau_{d}$. In general, the field
coefficients defined in equation (3) are complex functions, whose real part
gives the response function of the material: in this way, we can glean
information about their resonant behavior by treating them as rational
functions of the complex variable $\tilde{\omega}$. Poles of the
thermoelectric coefficients, known as cyclotron resonances, correspond to
values of $\omega$ for which equations (1) and (2) are degenerate, i.e. non-
invertable. This condition is equivalent to the vanishing of a determinant,
and means each response function has the same pole structure. As such, the
resonances are the four solutions to the equation

$q(\omega_{p})^{2}=-\left({\frac{\delta n}{n}\omega_{c}\tau_{0}}\right)^{2},$
(8)

which are

$\omega_{p}=-\frac{i}{\tau_{d}}-\frac{i}{2\tau_{0}}\pm_{2}i\sqrt{\frac{1}{4\tau_{0}^{2}}-\omega_{c}^{2}\mp_{1}i\frac{\delta
n}{n}\frac{\omega_{c}}{\tau_{0}}},$ (9)

with signs $\pm_{1}$ and $\pm_{2}$ uncorrelated. So long as the pole frequency
is close to the real axis in the complex plane, the location of the maxima of
the thermo-electric coefficients will be well approximated by
$\operatorname{Re}[\omega_{p}].$ We are therefore led to study the real and
imaginary parts of $\omega_{p}$, which are

$\displaystyle
2\tau_{0}\operatorname{Re}[\omega_{p}]=\begin{cases}\displaystyle\pm_{1-2}A\sin\left({\frac{1}{2}\tan^{-1}x}\right)&1>z\\\
\\\
\displaystyle\pm_{2}A\cos\left({\frac{1}{2}\tan^{-1}x}\right)&1<z,\end{cases}$
(10)

and

$\displaystyle 2$ $\displaystyle\tau_{0}\operatorname{Im}[\omega_{p}]=$
$\displaystyle\begin{cases}\displaystyle-(1+2\Gamma_{d})\mp_{2}A\cos\left({\frac{1}{2}\tan^{-1}x}\right)&1>z\\\
\\\
\displaystyle-(1+2\Gamma_{d})\pm_{2-1}A\sin\left({\frac{1}{2}\tan^{-1}x}\right)&1<z,\end{cases}$
(11)

for $\pm_{1-2}=(\pm_{1})(\mp_{2})$, $z:=2\tau_{0}\omega_{c}$, $x:=(2z\delta
n/n)(1-z^{2})$, $A=\left({(1-z^{2})^{2}+\left({2(\delta
n/n)z}\right)^{2}}\right)^{1/4}$, and $\Gamma_{d}:=\tau_{0}/\tau_{d}$. Notice
that the ambiguous sign in front of each expression implies that cyclotron
resonances come in pairs: if $\omega$ is a resonant frequency, then $-\omega$
is as well. Experimentally, this means the behavior of the maxima of any
thermoelectric conductivity experiment will be determined by the continuous
function $|\operatorname{Re}[\omega_{p}]|$, as the temperature, chemical
potential, and external magnetic field are varied; see Fig. 1 for a plot of
this function across both regimes. We remark that $\delta n/n=1$ describes the
case in which there are only electrons and no holes, in whci case the
cyclotron frequency reduces to the ordinary Fermi liquid result
$\omega_{p}=eB/m^{*}$. Equation (10) tells us the approximate location of the
peak of the hydrodynamic cyclotron resonance, but it does not give us
information about the width of that peak, or the accuracy of the
approximation; if we want our experimental signals to be sharp, we must look
for resonances which have a small imaginary part, i.e. for which
$\operatorname{Im}[\omega_{p}]/\operatorname{Re}[\omega_{p}]\ll 1$.

Figure 1: Positive real part of the cyclotron resonance frequency, measured in
units of $2\tau_{0}$. Curves for negative $\delta n/n$ are omitted, as they
are identical to positive curves of the same magnitude in $\delta n/n$.

Unlike for $\operatorname{Re}[\omega_{p}]$, the magnitude of
$\operatorname{Im}[\omega_{p}]$ depends on our choice of the signs $\pm_{1}$
and $\pm_{2}$. Assuming that the resonance behavior is dominated by the
closest pole to the real axis with positive real part, we are free to take
$|\operatorname{Im}[\omega_{p}]|$ to mean the smallest (in magnitude) positive
value of the right-hand side of (II). A plot of this function for
$\Gamma_{d}=0$, i.e. in the perfectly clean limit, is included in Figure (2).

Figure 2: Distance from the real axis of the closest hydrodynamic cyclotron
resonance with positive real part, measured in units of $2\tau_{0}$.

Comparing Figures 1 and 2 qualitatively, we see that near
$\omega_{c}\tau_{0}=1/2$, the resonant frequency is maximally displaced from
the Fermi liquid result of $\omega_{p}=\omega_{c}$ for small $\delta n/n$, but
that this region also corresponds to rapid enhancement in the magnitude of
$|\operatorname{Im}[\omega_{p}]|$. To gauge the quality of the resonance peak
at given values of $z$ and $\delta n/n$, we introduce a function
$\Gamma:=|\operatorname{Im}[\omega_{p}]|/|\operatorname{Re}[\omega_{p}]|$.
Broadly speaking, $\Gamma$ measures the width of the resonance, normalized by
the magnitude of its frequency: an ideal resonance corresponds to $\Gamma=0$,
while $\Gamma\gg 1$ implies a response function which is not determined by
resonant behavior. At $\delta n/n=0.1$, we have
$\Gamma:=|\operatorname{Im}[\omega_{p}]|/|\operatorname{Re}[\omega_{p}]|\approx
2.16$, corresponding to a broad peak. A heat map of the quality function
$\Gamma=\Gamma(z,\delta n/n)$ is presented in Figure 3(a): notably, the signal
of a hydrodynamic cyclotron resonance is weakest in exactly the region where
the effect of hydrodynamics on the resonance is strongest, i.e. where
$\Delta\omega_{c}:=|\operatorname{Re}[\omega_{p}]-\omega_{c}|$ is maximized,
as can be seen from Figure 3(b). In order to determine the optimal region of
phase space for observing hydrodynamics, we will maximize $\Delta\omega_{c}$,
subject to a constraint placed on $\Gamma$, as we will explore numerically in
the following section.

Figure 3: Heat maps for the quality function $\Gamma$ (above), and the
difference between the hydrodynamic cyclotron resonance and the Fermi liquid
value $\Delta\omega_{c}$ (below), both as functions of $z$ and $\delta n/n$.
The red curve on each plot indicates the $\Gamma=0.5$ level-set along which
$\Delta\omega_{c}$ is maximized, as discussed in the text.

In addition to clean system limit, we can also consider the fully compensated,
i.e. $\delta n/n$ limit, which occurs at zero chemical potential, and
corresponds to an equal density of electrons and holes. For such systems, the
equations for the thermo-electric coefficients simplify greatly: for example,
the longitudinal conductivity becomes

$\sigma_{xx}(\tilde{\omega})=\sigma_{D}\frac{\tilde{\omega}}{q(\tilde{\omega})}.$
(12)

Recalling that the conductivity measured in the laboratory is the real part of
equation (12), we find that the frequency which maximizes the conductivity is
given by

$\displaystyle\left({\omega_{\sigma}^{xx}}\right)^{2}$
$\displaystyle=\frac{(1+2\Gamma_{d})\omega_{c}\sqrt{q\left({\Gamma_{d}}\right)}}{(1+\Gamma_{d})\tau_{0}}-\frac{\Gamma_{d}q\left({\Gamma_{d}}\right)}{(1+\Gamma_{d})\tau_{0}^{2}},$
(13)

and that this corresponds to a maximal value of

$\frac{\left({\sigma_{xx}}\right)_{\text{max}}}{\sigma_{D}}=\frac{1+\Gamma_{d}-2\tau_{0}\omega_{c}\left({\tau_{0}\omega_{c}+\sqrt{q\left({\Gamma_{d}}\right)}}\right)}{(1+2\Gamma_{d})\left({1-(2\tau_{0}\omega_{c})^{2}}\right)}.$
(14)

Since $\Gamma_{d}$ and $\tau_{0}\omega_{c}$ are always positive, this equation
is bounded above by 1, and achieves this value if and only if $\Gamma_{d}=0$.
$\sigma_{D}$ therefore represents the ideal limit for hydrodynamic
conductivity at charge neutrality in bilayer graphene.

To conclude this section, we remark that similar calculations can be preformed
for the other thermo-electric coefficients, but that at zero chemical
potential both particles and holes carry the same entropy so that thermal and
electric transport look quite similar.

##  III Numerical computation

We consider the case of a symmetric two-fluid plasma with
$m_{e}^{*}=m_{h}^{*}=0.033\cdot m_{e}$, and for experimentally relevant
temperature $T=90$ K. Values of $B=0.18$T, and $\mu=0.66/\beta$ were chosen to
maximize $\Delta\omega_{c}$ subject to the condition $\Gamma=0.5$—see Figure
(3). Model parameters, such as $\tau_{0}$ and $\tau_{d}$, are computed in
accordance with the supplementary material of [1]; using equation (10), we
predict a hydrodynamic cyclotron frequency of
$\operatorname{Re}[\omega_{p}]=0.67\cdot\omega_{c}=0.59$ THz. Figure 4
displays the longitudinal electrical conductivity, $\sigma_{xx}$, of the
symmetric two-fluid system as a function of the drive frequency $\omega$: the
exact maximum in the conductivity occurs at
$\omega_{\text{max}}=0.79\cdot\omega_{c}=0.70$ THz, corresponding to a
displacement $\Delta\omega_{c}=0.21\cdot\omega_{c}=1.84\times 10^{11}$ Hz from
the conventional cyclotron resonance. We likewise generated plots for oher
thermoelectric coefficients, such as the Hall resistance and the Seebeck
coefficient, (3) using the same parameters as above, however their resonance
peaks, being principally determined by the pole structure of $\tilde{\omega}$,
demonstrate no new behaviour, and are not included here.

Figure 4: Longitudinal electrical conductivity, $\sigma_{xx}$, plotted in
units of $\sigma_{0}=e^{2}/h$. A value of $T=90$K, $\mu=0.66/\beta$, and
$B=0.18$T, were chosen for the temperature, chemical potential, and magnetic
field respectively. The dashed vertical line represents the classical Fermi
liquid cyclotron resonance $\omega_{c}=eB/m*$, while the red and blue lines
are the numerically and theoretically computed values of the hydrodynamic
cyclotron resonance $\omega_{p}$ respectively.

##  IV Discussion

We begin this section with a disclaimer: the results given above pertain to
homogeneous response in the linear-response regime, and so do not strictly
speaking depend on the hydrodynamics of the underlying equations of motion. In
fact, it is not clear a priori that the hydrodynamic assumption made in [1],
i.e. that the electron and hole fluids equilibriate independently before
interacting, holds: however, the authors support their assumption with
convincing numerical data, and so we are confident that results derived from
these hydrodynamic equations will be valid, if only approximate.

Generically, non-interacting electron fluids in the presence of disorder have
transport properties given, up to constants, by the Drude model[39]. In
particular, Kohn’s theorem ensures that in isolation, such fluids must have a
resonance at exactly $\omega_{c}$. The physics becomes richer if a second band
is introduced, as long as the two bands are allowed to interact, e.g. via the
coulomb force. Under such conditions, each fluid drags on the other,
potentially leading to a retardation in the resonant frequency from the free
electron value of $\omega_{c}$. However, the same drag force which causes a
deviation in resonance frequency is also responsible for the broadening of the
signal; this effect is reflected in figures 1 and 2, which show that the the
region where $\omega_{c}$ is smaller than $\tau_{0}^{-1}$ but non-zero
corresponds to non-linear dependence of the real part of resonance on the
magnetic field, as well as rapid growth in the imaginary part of the
resonance.

Figures 1 and 2 also reveal the importance of charge compensation to the
hydrodynamic effects on the cyclotron resonance. Intuitively, this result is a
reflection of a broader principle: whenever a single time scale–coulomb,
disorder, magnetic, or anything else–becomes by far the shortest scale in the
problem, the picture of interacting fluids breaks down, and we recover
traditional Drude transport. Consequently, when the system is far from charge
compensation, the drag term of the dominant fluid will be suppressed by a
factor of $n^{(e/h)}/n$, meaning the non-linearity of the resonance will
vanish, as we see in figure 1; this corresponds to the case of a single fluid
interacting weakly with a background through the coulomb force.

Finally, we can see this principle at work in equation (12): as mentioned in
that section, $\sigma_{xx}$ is bounded above by $\sigma_{D}=ne\tau_{0}/m$,
which is the Drude conductivity except with the usual disorder timescale
$\tau_{d}$ replaced by the coulomb drag $\tau_{0}$, and that this limit is
achieved exactly when $\Gamma_{d}=0$. Furthermore its straightforward to check
that $\omega_{\sigma}^{xx}$, defined in equation (13), reduces to $\omega_{c}$
in the limit $\Gamma_{d}\rightarrow 0$, as we would expect. We then see that
the unusual behavior of the hydrodynamic resonance in two-fluid hydrodynamic
bilayer graphene is a consequence of a competition between interaction time
scales, and which interpolates between limits of Drude transport whenever one
single time scale is allowed to dominate.

##  V Conclusion

In this paper, we found analytic formulae for the thermoelectric coefficients
in the two-fluid hydrodynamic regime of bilayer graphene, starting from the
fluid dynamic equations (1) and (2). Our results indicate a departure from the
traditional cyclotron frequency $eB/m^{*}$ predicted by Kohn’s theorem,
revealing instead a characteristic dependence on the magnetic field and
carrier densities in the hydrodynamic regime of bilayer graphene. Our work
also suggests target conditions for the direct observation of a hydrodynamic
resonance in real bilayer graphene systems.

###### Acknowledgements.

JRC and AS gratefully acknowledge support under NSF Grant No. DMR-2029401. EH
acknowledges support under NSF CAREER DMR-1945278. G.V. was supported by the
Ministry of Education, Singapore, under its Research Centre of Excellence
award to the Institute for Functional Intelligent Materials (I-FIM, project
No. EDUNC-33-18-279-V12)

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