Plasmons, the collective oscillations of interacting electrons, possess
emergent properties that dramatically alter the optical response of metals. We
predict the existence of a new class of plasmons -- chiral Berry plasmons
(CBPs) -- for a wide range of two-dimensional metallic systems including gapped
Dirac materials. As we show, in these materials the interplay between Berry
curvature and electron-electron interactions yields chiral plasmonic modes at
zero magnetic field. The CBP modes are confined to system boundaries, even in
the absence of topological edge states, with chirality manifested in split
energy dispersions for oppositely directed plasmon waves. We unveil a rich CBP
phenomenology and propose setups for realizing them, including in anomalous
Hall metals and optically-pumped 2D Dirac materials. Realization of CBPs will
offer a new paradigm for magnetic field-free, sub-wavelength optical
non-reciprocity, in the mid IR-THz range, with tunable splittings as large as
tens of THz, as well as sensitive all-optical diagnostics of topological bands.Comment: 10 pgs, 3 fg

Rudner, Mark S.

Song, Justin C. W.

English

arXiv

Copenhagen University Research Information System

Chiral plasmons without magnetic field

Plasmons, the collective oscillations of interacting electrons, possess emergent properties that dramatically alter the optical response of metals. We predict the existence of a new class of plasmons—chiral Berry plasmons (CBPs)—for a wide range of 2D metallic systems including gapped Dirac materials. As we show, in these materials the interplay between Berry curvature and electron–electron interactions yields chiral plasmonic modes at zero magnetic field. The CBP modes are confined to system boundaries, even in the absence of topological edge states, with chirality manifested in split energy dispersions for oppositely directed plasmon waves. We unveil a rich CBP phenomenology and propose setups for realizing them, including in anomalous Hall metals and optically pumped 2D Dirac materials. Realization of CBPs will offer a powerful paradigm for magnetic field-free, subwavelength optical nonreciprocity, in the mid-IR to terahertz range, with tunable splittings as large as tens of THz, as well as sensitive all-optical diagnostics of topological bands

Caltech Authors - Main

Supplementary Online Information for “Chiral plasmons without magnetic field”
Justin C. W. Song1 and Mark S. Rudner2
1 Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125 USA and
2 Center for Quantum Devices and Niels Bohr International Academy,
Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
Current density and the conductivity tensor
For a complementary view to the approach employed
in the main text, here we present an alternative formu-
lation of the plasmon equations of motion. Rather than
focusing on the semiclassical equations of motion for the
particle and velocity density fields, here we work with
the charge density ρ(r, t) = −en(r, t), the current density
j(r, t), and the conductivity tensor σ. As we will be in-
terested in oscillatory collective (plasmon) modes, it will
be useful to work in terms of Fourier modes, ρ(r, t) =
ρω(r)e
iωt, j(r, t) = jω(r)e
iωt, and φ(r, t) = φω(r)e
iωt.
The charge and current densities obey a continuity equa-
tion, which in terms of the Fourier modes reads:
iωρω(r) +∇ · jω(r) = 0, jω(r) = σ(ω)∇φω(r). (S-1)
Importantly, even at B = 0, the conductivity σ(ω) pos-
sesses off-diagonal contributions, captured by the anoma-
lous Hall conductivity σAHxy .
In terms of frequency dependence, the longitudinal
conductivity takes a standard Drude form, while the
Hall conductivity, arising from the anomalous velocity
1
h¯∇φ×Ω(p) [1], is frequency-independent:
σxx = σyy =
n0e
2/m
iω˜
, σAHxy = const. (S-2)
Here ω˜ = ω − i/τtr, where τtr is a transport scattering
time that arises, e.g., from impurity scattering. We con-
sider collective modes in the regime ωτtr  1, where the
1/τtr contribution to σxx can be neglected.
The anomalous Hall conductivity is easily connected
with quantities in the main text via σAHxy = Fe2/h. Here
F = ∑i ∫ d2p Ωi(p)f0i (p)/(2pih¯)2 is the dimensionless
Berry flux, with f0i (p) the equilibrium band occupancy,
and Ω(p) the Berry curvature. Throughout this work we
only consider terms up to linear order in ∇φ.
In this formulation, chiral Berry plasmons (CBPs) at
system edges or in disks arise in exactly the same way
as shown in the main text, with v¯(r, t) and F traded for
j(r, t) and σAHxy . That is, Eqs. (S-1) and (S-2) of this
supplement, with 1/τtr → 0, are equivalent to Eqs. (1)
and (2) of the main text; following the same steps of
derivation yields Eqs. (4-11) of the main text with the
replacement F = hσAHxy /e2.
We note parenthetically that Eq. (S-2) is different from
what would be obtained in the analogous case for magne-
toplasmons in a system subjected to an applied magnetic
field. While both cases feature a nonzero Hall conduc-
tivity σxy, in the case of magnetoplasmons the applied
magnetic field induces a Lorentz force that affects elec-
tronic motion by modifying the force balance. This leads
to different forms of σxx and σxy from those in Eq. (S-
2); in particular, σxy in the presence of a finite mag-
netic field exhibits an ω dependence [2], in contrast to
Eq. (S-2). This yields qualitatively different behavior of
collective modes. For example, bulk magnetoplasmons
are gapped, whereas bulk plasmons in the presence of a
Berry flux giving the same dc-Hall conductivity are gap-
less (see discussion in main text).
CBP dipole mode in a disk
Here we present a more complete analysis of the CBP
dipole modes in a disk, which dominate the optical ab-
sorption in metallic disks. These dipole modes can
be conveniently described through the center of mass
(COM) motion, {x(t)}, wherein all internal forces cancel
(viz. Newton’s third law). Here {·} denotes the COM
average, with {x(t)} = ∫ d2xn(x, t)x, and {p(t)} =∫
d2x p¯(x, t). The COM equations of motion can be ob-
tained from Eq. (1) of the main text via integration by
parts, along with the condition that the velocity vanishes
when |x| → ∞. This yields
∂t{x} =
{
∂ε
∂p
}
+
1
h¯
{e∇φ×Ω(p)}, ∂t{p} = {e∇φ},
(S-3)
where E is a self-generated electric field associated with
the plasmon motion.
To capture the dipolar mode, we use the harmonic po-
tential −eφ(x) = m2 ω20 |x|2 for the electrons (see e.g., Ref.
[3]) where ω0 is the bare plasmon frequency in a disk
(with diameter d) in the absence of Berry curvature.
In analyzing the COM equations of motion, we first
note that e∇φ(x) depends only on x, and Ω(p) only on p.
Keeping terms to linear order in δn, we find the anoma-
lous velocity contribution {va} for the COM as
{e∇φ×Ω} =
∑
x
[
e∇φ(x)×
∑
i
∫
d2p
(2pih¯)2
Ω(p)fi(x,p, t)
]
=
[∑
x
e∇φ(x)F(x, t)
]
× zˆ ≈ ζ{x(t)} × zˆ,
(S-4)
2where the Berry flux is given by F(x, t) ≡∑
i
∫
d2p Ωi(p)fi(x,p, t)/(2pih¯)
2, the constant ζ can be
obtained as detailed below, and i is a band index. Here
we will concentrate on the motion of bulk electrons in
a given band, n. We therefore take fi<n = 1 every-
where inside the disk for bands below n (but vanishing
outside the disk). For the purpose of estimating param-
eters, we make an assumption of local equilibrium and
set fn(x,p, t) = [e
β(εnp−µ(x,t)) + 1]−1, where µ(x, t) is a
space and time varying chemical potential. Adopting a
simple model of a rigidly moving disk of charge with con-
stant density n0, and density equal to zero outside, gives
F = Fn(x, t)/n0 and ζ = − Fn0mω20 . Here F is obtained
with a fixed uniform chemical potential µ.
Using {e∇φ} = −mω20{x(t)}, Eq. (S-4), and substitut-
ing into Eq. (S-3), we obtain the equations of motion for
the dipole mode: (∂2t +Aij){xj} = 0, where
A =
(
ω20 ωa∂t
−ωa∂t ω20
)
, ωa =
Fω20m
n0h¯
. (S-5)
Here we have used
{
∂ε
∂p
}
= 1m{p}+O(δn2) and kept only
terms up to linear order in δn.
Writing {x} = x0eiωt, we obtain a secular equation
Mx0 = 0 where M = −ω2 + A, with ∂t replaced by
iω within A. Plasmons are given by the zero modes,
det(M) = 0, yielding the split dispersion relation
ω± =
√
ω20 +
ω2a
4
± ωa
2
, (S-6)
where ω±, ω0 > 0. (In the following we will use only posi-
tive frequencies; a similar analysis yields the same results
for negative ω±, ω0 as well). These plasmon modes are
chiral as they correspond to a rotating COM displace-
ment and momentum
{x(t)}± = |x0|√
2
(
1
±i
)
eiω±t, {p(t)}± = iω
2
0m
ω±
{x(t)},
(S-7)
where {p(t)}± and {x(t)}± are offset by a phase of pi/2.
As a result, a rotating {x(t)}± gives rise to a circulat-
ing momentum/current density (see Fig. 3a of the main
text). The clockwise and anticlockwise (ω+ and ω−) mo-
tions of chiral plasmons sketched in Fig. 3a are oriented
for a positive Berry flux, F , pointing in the zˆ direction.
This direction sets the orientation of ωa. For opposite
sign of F the orientations are switched. The distinct
frequencies ω± arise from the anomalous velocity {va}
adding/subtracting propagation speed from the modes
at F = 0.
The CBP dipole mode can manifest in distinct split
peaks for optical absorption. This splitting can be ana-
lyzed by writing the current density as j = en0∂t{x} by
inverting M above, and relating current to electric field
via j = gE, where E is probing field, and g is the con-
ductivity tensor. Optical absorption is characterized by
the real part of the longitudinal conductivity, gxx, as
Re[gxx(ω)] =
1
2
∑
±
DΓω2
(ω2 ± ωωa − ω20)2 + Γ2ω2
, (S-8)
where D = n0e2m is the Drude weight, and Γ is the
transport relaxation rate, included phenomenologically
via ∂2t → ∂2t + Γ∂t. This yields the split peak optical
absorption for the disk geometry shown in Fig. 3b of the
main text.
Optical Selection rules for gapped Dirac materials
Here we detail how circularly polarized light can yield
valley selectivity in gapped Dirac materials (GDMs)
where inversion symmetry has been broken. This selec-
tivity arises due to the pseudo-spinor nature of the wave-
functions on the A/B sublattices. The selection rules
were derived in Ref. [4] by analyzing the orbital mag-
netic moments of electronic wavepackets in the K and
K ′ valleys. For the reader’s convenience, here we present
an alternative calculation based on Fermi’s golden rule.
The low energy Hamiltonian for gapped Dirac materi-
als can be described as H = HK +HK′ , where
HK = vσ+ · pK + ∆σz, HK′ = vσ− · pK′ + ∆σz,
(S-9)
where HK , HK′ describe electrons close to the K and K
′
points, v is the Fermi velocity, and σ± = σxxˆ±σyyˆ, with
σx,y,z the Pauli matrices. Here pK,K′ describe momenta
taken relative to the K and the K ′ points, respectively,
and 2∆ is the gap size. In the following we shall drop
the explicit K and K ′ labels on pK,K′ for brevity.
A variety of systems obey Eq. (S-9), including van der
Waals heterostructures where A/B sublattice symmetry
has been broken as in G/hBN heterostructures. In such
systems the gap size 2∆ corresponds to the asymmetry
of the potential on the A/B sublattices. Below we will
focus on this case for concreteness. However, the un-
derlying physics is general and applies to a broad range
of GDMs, such as dual-gated bilayer graphene and tran-
sition metal dichalcogenides, where inversion symmetry
has been broken.
The eigenfunctions of Eq. (S-9) can be expressed as
pseudo-spinors
|+〉K(K′) =
(
cos θ2e
−iτφ
sin θ2
)
, |−〉K(K′) =
(
sin θ2e
−iτφ
−cos θ2
)
,
(S-10)
where |±〉 denote states in the conduction (valence) band,
τ = 1 for valley K and τ = −1 for valley K ′, tan θ =
v|p|/∆, and tan φ = py/px. Here the energy eigenvalues
are ε±p = ±
√
v2|p|2 + ∆2.
We proceed by noting that when light with frequency
h¯ω ≥ 2∆ is incident on the GDM, electrons in the valence
3band can be excited into the conduction band. The light-
matter coupling is captured by writing p → p − eA/c,
where the vector potential A is related to the incident
light electric field E via A = icωE. Here E = E0e
iωt for
light with frequency ω. The rate of electron-hole pair
creation (absorption of photons) can be calculated via
Fermi’s golden rule
WK(K′) =
2pi
h¯
∑
p
|MK(K′)p |2δ(ε+p − h¯ω/2), (S-11)
with the matrix elements
MKk =
iev
ω
〈+|Exσx + Eyσy|−〉K
MK
′
k =
iev
ω
〈+|Exσx − Eyσy|−〉K′ . (S-12)
We note that the different signs in front of Ey in M
K
k and
MK
′
k arise from the different way A couples to pseudospin
in HK and HK′ (see Eq. S-9).
With the help of the identities
〈+|σx|−〉K(K′) = sin2 θ2e−iτφ − cos2 θ2eiτφ (S-13)
〈+|σy|−〉K(K′) = i
(
sin2 θ2e
−iτφ + cos2 θ2e
iτφ
)
,
and writing E0 = |E0| (xˆ+ iηyˆ)/
√
2 for left-handed (LH,
η = 1) and right-handed (RH, η = −1) circularly polar-
ized light, we obtain
W ηK = W0
[
2∆
h¯ω
+ η
]2
, W ηK′ = W0
[
2∆
h¯ω
− η
]2
. (S-14)
Here W0 = Nse
2|E0|2/(16h¯2ω), where Ns is the spin de-
generacy.
Equation (S-14) shows that electron-hole transitions in
the valleys K/K ′ can be selectively excited using LH/RH
circularly polarized light (see Fig. 3c inset of the main
text). Indeed for h¯ω = 2∆, perfect selection of K or K ′
electron-hole transitions can be achieved, in agreement
with Ref. [4]. We emphasize that this selectivity comes
from the orbital physics of light-matter coupling; it does
not require or involve spin-orbit coupling and can even
arise in materials with negligible spin orbit-coupling as
modeled by Eq. (S-9).
Optically pumped “On-Demand” CBP dipole mode
in gapped Dirac materials
Here we consider how CBPs might arise in non-
magnetic gapped Dirac materials such as MoS2, or
gapped G/h-BN. To analyze this, we focus on nomi-
nally time reversally invariant gapped Dirac materials.
We model the valley dependent F as [5]: FK,K′ =
τzNs
n˜1/2
2(nK,K′+n˜)1/2
sgn ∆, where nK,K′ are the valley car-
rier densities, τz = ∓ for the K,K ′ valleys, Ns = 2 is the
spin degeneracy, and n˜ = ∆2/4piv2F h¯
2 gives a characteris-
tic density scale. The bandgap is 2|∆|. When nK = nK′ ,
the total flux F = FK + F ′K vanishes.
However, a non-vanishing net Berry flux Fpe for the
photoexcited system is achieved when nK 6= nK′ (Fig. 3c
inset of the main text). Setting nK′ = 0, giving FK′ =
+1, the valley polarized electron (nel) and hole (nh) pop-
ulations concentrated at the K valley conduction and va-
lence band extrema yield
Fpe ≈ 1− 1
2
[ n˜1/2√
n˜+ nh
+
n˜1/2√
n˜+ nel
]
,
n˜ = 1.8× 1013 (∆[eV])
2
(vF [cm s−1]/108)2
cm−2. (S-15)
Here we have used that Berry curvature (including its
sign) is the same for electrons and holes, following from
particle-hole symmetry of the Dirac Hamiltonian, and
used the convention nel,h > 0. In deriving Eq. (S-15), we
used the spin degeneracy Ns = 2 for the K
′ valley, and
noted that only a single spin species in the K valley is
excited by the circularly polarized light.
For demonstration we examine CBPs in the disk ge-
ometry as above. We analyze the coupled motion of pho-
toexcited electrons and holes in a single valley via COM
coordinates ({xel}) and ({xh}), respectively, giving:
[
∂2t +
(
Ael −Ael
−Ah Ah
)]( {xel}
{xh}
)
= 0, (S-16)
where Ael and Ah are defined as in Eq. (7) of the main
text, with the density n0 and plasmon mass m replaced
by the appropriate values for electrons or holes. In writ-
ing these equations, we have used that the restoring force
arising from the mutual attraction of photoexcited elec-
tron and hole subsystems is {e∇φ}el(h) ≈ ∓α({xel} −
{xh}), where {·}el,h denote the COM averages for elec-
tron and hole distributions and the upper (lower) sign is
for electrons (holes). Here α characterizes the strength
of the electron-hole interaction. For brevity, in the fol-
lowing analysis we set Ael = Ah, giving α = mω
2
0 , where
ω0 is the plasmon frequency associated with a unipolar
system with carrier density n = nel = nh.
Zero modes of Eq. (S-16) with ∂2t → −ω2 yield CBPs
with {xel} = −{xh}, giving:
ωel−h± =
√
ω2a + 2ω
2
0 ± ωa, (S-17)
where ωa is given by Eq. (7) of the main text with F re-
placed by Fpe. Importantly, ωel−h± chiral (electron-hole)
plasmons feature co-rotating {xel} and {xh}. This dipole
like rotation follows from the equal sign of F for electrons
and holes. Modes with {xel} = {xh} are also eigenmodes
of Eq. (S-16). However, they have frequency, ω = 0.
4[1] Xiao D., Meng M. C., Niu Q., Berry phase effects on elec-
tronic properties, Rev. Mod. Phys. 82, 1959-2007 (2010).
[2] Fetter A. L., Edge magnetoplasmons in a two-dimensional
electron fluid confined to a half-plane, Phys. Rev. B 33,
3717 (1986).
[3] Yan H., Li Z., Li X., Zhu W., Avouris P., Xia F., In-
frared Spectroscopy of Tunable Dirac Terahertz Magneto-
Plasmons in Graphene, Nano Lett. 12, 3766 (2012).
[4] Yao W, Xiao D, Niu Q (2008) Valley-dependent optoelec-
tronics from inversion symmetry breaking, Phys. Rev. B
77, 235406.
[5] Lensky Y. D., Song J. C. W., Samutpraphoot P., Levi-
tov L. S., Topological Valley Currents in Gapped Dirac
Materials, Phys. Rev. Lett. 114, 256601 (2015).


Chiral plasmons without magnetic field

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