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Title
Polaritons in van der Waals materials.
Permalink
https://escholarship.org/uc/item/1jj1t706
Journal
Science (New York, N.Y.), 354(6309)
ISSN
0036-8075
Authors
Basov, DN
Fogler, MM
García de Abajo, FJ
Publication Date
2016-10-01
DOI
10.1126/science.aag1992
Peer reviewed
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REVIEW SUMMARY
◥
QUANTUM MATERIALS
Polaritons in van der Waals materials
D. N. Basov,* M. M. Fogler, F. J. García de Abajo
BACKGROUND: Light trapped at the nanoscale,
deep below the optical wavelength, exhibits an
increase in the associated electric field strength,
which results in enhanced light-matter interac-
tion. This leads to strong nonline ar it i es , lar g e
photonic forces, and enhanced emission and
absorption probabilities. A practical approach
toward nanoscale light trapping and manipula-
tion is offered by interfaces separating media
with permittivities of opposite signs. Such inter-
faces sustain hybrid light-matter modes
involving collective oscillations of po-
larization charges in matter , hence
the term polaritons. Surface plasmon
polaritons, supported by electrons in
metals, constitute a most-studied prom-
inent example. Yet there are many
other varieties of polaritons, including those
formed by atomic vibrations in polar insulators,
excitons in semiconductors, Cooper pairs in
superconductors, and spin resonances in (anti)
ferromagnets. Together, they span a broad re-
gion of the electromagnetic spectrum, ranging
from microwave to ultraviolet wavelengths. We
discuss polaritons in van der Waals (vdW) ma-
terials: layered systems in which individual atomic
planes are bonded by weak vdW attraction (see
the figure). This class of quantum materials in-
cludes graphene and other two-dimensional
crystals. In artificial structures assembled from
dissimilar vdW atomic layers, polaritons asso-
ciated with different constituents can interact
to produce unique optical effects by design.
ADVANCES: vdW materials host a full suite of
different polaritoni c modes wi th the highest
degree of confinement among all known mate-
rials. Advanced near-field imaging methods
allow the polaritonic waves to be launched and
visualized as they travel along vdW
layers or through multila yered hetero-
structures. Spectroscopic and nano-
imaging experiments have identified
multiple routes toward manipulation
of nano-optical phenomena endowed
by polaritons. A virtue of polaritons in
vdW systems is their electrical tunability. Fur-
thermore, in heterostructures assembled from
dissimilar vdW layers, different brands of polar-
itonsinteractwitheachother,thusenablingun-
paralleled control of polaritonic response at the
level of single atomic planes. New optoelectronic
device concepts aimed at the detection, harvest-
ing, emission, propagation, and modulation of
light are becoming feasible as a result of com-
bined synthesis, nanofabrication, and modeling
of vdW systems. The extreme anisotropy of
vdW systems leading to opposite signs of the
in-plane and out-of-plane permittivities of the
same layered crys tal enables efficient polaritonic
waveguides, which are instrumental for subdif-
fractional focusing and imaging. In addition to
near-field optical probes facilitating nanoimaging,
coupling to polaritons can be accomplished via
electrical excitation and nonlinear wave mixing.
OUTLOOK: Potenti al outcomes of polariton
exploration in vdW heterostructures go beyond
nano-optical technologies. In particular, im-
ages of polaritonic standing and traveling waves
contain rich insights into quantum phenomena
occurring in the host material supporting po-
laritons. This line of inquiry into fundamental
physics through polaritonic observ ations con-
stitutes an approach toward optics-based ma-
terials research. In particular , the strong spatial
confinement exhibited by vdW polarit ons in-
volves large optical-field gradients—or equiva-
lently, large momenta—which allows regions
of the dispersion relations of electrons, phonons,
and other condensed-matter excitations to be
accessed beyond what is currently possible with
conventional optics. Additionally, polaritons
created by short and intense laser pulses add
femtosecond resolution to the study of these
phenomena. Alongside future advances in the
understanding of the physics and interactions
of vdW pol aritons, solutions to application chal-
lenges may be anticipated in areas such as loss
compensation, nanoscale lasing, quantum optics,
and nanomanipulation. Th e field of vdW polar-
itonics is ripe for exploring genuinely unique
physical scenarios and exploiting these new
phenomena in technology.
▪
RESEARCH
SCIENCE sciencemag.org 14 OCTOB ER 2016 • VOL 354 ISSUE 6309 195
The list of author affilia tions is availabl e in the full article online.
*Corresponding author. Email: db3056@columbia.edu
Cite this article as D. N. Basov et al., Scienc e 354, aa g1 992
(2016). DOI: 10.1126/science.aag1992
Polaritons in van der Waals (vdW) materials. Polaritons—a hybrid of light-matter oscillations—can originate in different physical phenomena: conduction
electr ons in graphene and topological insulators ( surface plasmon polaritons ), infr ared-activ e phonons in boron nitride (phonon polaritons ), ex citons in
dichalcogenide materials (e x citon po laritons), superfluidity in FeSe- and Cu-based superconductors with high critical temperature T
c
(Cooper-pair polaritons),
and magnetic resonances (magnon polaritons). The family of vdW materials supports all of these polaritons. The matter oscillation component result sin
negativ e permittivity (e
B
< 0) of the polaritonic material, giving rise to optical-field confinement at the interface with a positive-permittiv ity (e
A
> 0) environment .
vdW polaritons exhibit str ong confinement, as defined by the ratio of incident light wavelength l
0
to polariton wavelength l
p
.
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REVIEW
◥
QUANTUM MATERIALS
Polaritons in van der Waals materials
D. N. Basov,
1,2
* M. M. Fogler,
1
F. J. García de Abajo
3,4
van der Waals (vdW ) materials consist of individual atomic planes bonded by weak
vdW attraction. They display nearly all optical phenomena found in solids, including
plasmonic oscillations of free electrons characteristic of metals, light emission/lasing
and excitons encountered in semiconductors, and intense phonon resonances typical
of insulators. These phenomena are embodied in confined light-matter hybrid modes
termed polaritons—excitations of polarizable media, which are classified according
to the origin of the polarization. The most studied varieties are plasmon, phonon, and
exciton polaritons. In vdW materials, polaritons exhibit extraordinary properties that are
directly affected by dimensionality and topology, as revealed by state-of-the-art imaging of
polaritonic waves. vdW heterostructures provide unprecedented control over the
polaritonic response, enabling new quantum phenomena and nanophotonics applications.
A
tomically thin two-dimensional (2D) crys-
talline layers constitute the elemental build-
ing blocks of van der Waals (v dW) materials.
Exfoliated atomic layers are structurally
robust and amenable to assembly to pro-
duce complex heterostructures. These mater ials
support a variety of polaritons associated with
oscillations of conduction electrons, phonons, and
excitons, as well as their hybrids (e.g., plasmon-
phonon polaritons). A number of vdW mater ials
display extraordinary quantum phenomena: high–
critical temperature (T
c
) superconductivity, exotic
ma g n e tism , top o l ogically protected states, strong
Coulomb interactions, and non–Fermi-liquid be-
havior . All of these properties permeate the polar-
itonic response of vdW systems.
In the ongoing quest for exploration of polaritons,
scanning optical near-field imaging (Fig. 1) has
had an exceptional impact. This technique uses
the sharp tip of an atomic force microscope
(AFM) as an optical antenna (1, 2), allowing
one to detect how incident light of free-space
wavelength l
0
is scattered at the apex of the
tip in the proximity of the studied specimen
(Fig. 1A). The obtained signal is gover ned by the
local electric field of the polariton wave launched
by the tip, rendering nanometer spatial reso-
lution as the tip is raster-scanned over the sam -
ple. The principal characteristics of polaritons,
including wavelength l
p
, confinement ratio l
0
/
l
p
, and quality factor Q (Table 1), reveal that
vdW polaritons are simultaneously compact and
long-lived. The polariton wavelength can often
be tuned with various methods, of which elec-
trical gating is of paramount importance. These
characteristics render vdW polaritons comple-
mentary and sometimes superior to those ob-
served in more conventional materials (3, 4).
A major challenge of polariton imaging and
spectroscopy stems from the large momentum
mismatch with free-space photons. However, ex-
perimentalists are becoming increasingly adept
at overcoming this difficulty. Figure 2 displays
various coupling schemes. Coherent launchers
(Fig. 2, A to C) have relatively small coupling cross
sections, although they can be enhanced through
optical antennas, including AFM tips (Fig. 2C)
and metal bars or disks (5). Incoherent launchers
(Fig. 2, D to F) can reach order-unity efficiency;
in particular , electron beams (Fig. 2D) eventually
will enable an impressive combination of energy
and space resolution (6).
Primer on polaritons
Polariton dispersion in thin layers
When the sample thickness d is much smaller
than the wavelength l
p
of polaritons, only the in-
plane optical response of the material is impor-
tant. In this thin-film limit, one finds
l
p
¼
2p
k
p
¼ 4p
2
Im
s
we
a
; d << l
p
ð1Þ
where e
a
is the permittivity of the environment,
s isthein-planeconductivity,w is the frequency,
and k
p
is the in-plane polariton wave vector. The
field of the polariton wave decreases exponen-
tially away from the interface over a characteristic
distance ~l
p
/2p. It is common to describe s in
Eq. 1 as the sum
sðwÞ¼
i
p
S
f
w þ it
−1
f
þ
i
p
wS
b
w
2
− w
2
b
þ iwt
−1
f
(2)
The first (Drude) and the second (Lorentz) terms
represent the contribution of free (f) and bound
(b) charges, respectively. The latter can also ac-
count for optical phonons. Different vdW mate-
rials can be modeled with a suitable choice of
parameters in Eq. 2: the spectral weights S
f
and S
b
, the exciton/phonon frequency w
b
, and
the phenomenological relaxation times t
f
and
t
b
(related to Q in Table 1 by t = Q/w).
The spectral weights in Eq. 2, and therefore
the polariton wavelengths of vdW materials, are
oftentunable.Ingraphene,S
f
scales with the
Fermi energy E
F
according to S
f
~(e/ħ)
2
E
F
(where
e isthechargeontheelectronandħ is the Planck
constant divided by 2p)(7); the value of S
f
can be
controlled via electrical gating, doping, and photo-
excitation. In insulators, where S
b
~(e
2
/ħ) f w
b
,
the dimensionless parameter f scales linearly
with the number N of atomic layers. In particular ,
f
ph
~ N
ffiffiffiffiffiffiffiffiffiffiffiffi
m=M
p
<< 1 for optical phonons and f
ex
~
N(D/ea
ex
)
2
for excitons. Here, m, M, a
ex
,andD are
the electron mass, the atomic mass, the exciton
Bohr radius, and the exc iton transition dipole,
respectively. In superconductors, the total Drude
weight is constant but is split between normal- and
sup er-c urre nt components, with a relative weight
depending on temperature. When applied to
graphene, Eqs. 1 and 2 readily explain why the
surface plasmon polariton (SPP) confinement
ratio l
0
/l
p
=(e
a
/a)(ħw/2E
F
) >> 1 can be extra-
ordinarily high (8): l
0
/l
p
scales with the inverse
of the fine-structure constant a ≈ 1/137. How-
ever, a stronger confinement is accompanied by
larger damping rate t
−1
f
[which also increases
with w (9, 10)].
Polariton dispersion in slabs
and heterostructures
For highly confined polaritons (or thicker sam-
ples), the condition l
p
>> d may not hold, so the
polariton dispersion becomes more intricate
(Fig. 3). Both the in- and out-of-plan e responses
need to be considered, and it is more conve-
nient to use the permittivity tensor, whose in-
plane component e
∥
relates to s(w)ase
∥
=1+
(4pis/wd), whereas the out-of-plane component
e
⊥
may dif f e r from e
∥
in both magnitude and sign
be c ause of the strong anisotropy of vdW mate-
rials (Fig. 3, E and F). Additional complications
arise if the sample is a heterostructure made o f
dissimilar vdW materials (metals, insulators, or
semiconductors). These more complex dispersions
can be d issected into simpler elements (11, 12)
as in Fig. 3, A to G.
SPPs are supported by materials that possess
mobile charges: metals, doped semiconductors,
and superconductors. We find three types of elec-
tromagnetic modes in these materials (Fig. 3A):
two in the bulk (photon and plasmon) and one
confined at the surface (lower curve, representing
theTM-polarizedSPP).Thetransverseupperbranch
also starts at frequency w
p
and disperses upward at
higher wave vector k. This behavior results from
l evel repulsion between the photon (dashed line
in Fig. 3A) and the zero-frequency (Drude) res-
onance of the conductor . The high-k SPP is often
referred to as a surface plasmon (SP). Its disper-
sion asymptotically approaches w
SP
= w
p
=
ffiffiffi
2
p
.
The SPPs at the surfaces of a thin conduct-
ing film of thickness d << c/w
p
split into two
branches of opposite symmetry. The lower,
RESEARCH
SCIENCE sciencemag.org 14 OCT OBER 2016 • VOL 354 ISSUE 6309 aag1992-1
1
Department of Physics, University of California, San Diego,
CA, USA.
2
Department of Physics, Columbia University, New
York, NY, USA.
3
Institut de Ciencies Fotoniques, Barcelona
Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain.
4
Institució Catalana de Recerca i Estudis
Avançats, 08010 Barcelona, Spain.
*Corresponding author. Email: db3056@columbia.edu
aag1992-2 14 OCTOBER 2016 • VOL 354 ISSUE 6309 sciencemag.org SC IENCE
Table 1. Characteristics of polaritons in vdW materials. Tunability methods marked with asterisks indicate already demonstrated results. All experimental
entries are obtained under ambient conditions, except for Cooper-pair plasmons at T = 5 K. SPP, surface plasmon polariton; DE, dielectric environment; N/A,
not available.
Polariton types
(materials)
Image Energy range (meV)
l
p
(nm)
max
l
0
/l
p
max Q Tunability methods References
Dirac SPP (graphene) Fig. 1A <1000 50 to 450 220 40 Gating,* doping,*
photoexcitation,* DE*
(10, 20, 21, 100)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Edge SPP (graphene) Fig. 1B <1000 50 to 200 200 10 Gating,* doping,
photoexcitation, DE*
(64, 65)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
One-dimensional SSP
(carbon nanotubes)
Fig. 1C <200 100 to 1000 26 Conducting channels,* DE (30)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Superlattice SPP
(graphene/hBN moiré
superlattices)
Fig. 1D <1000 50 to 250 220 4 Gating, doping,*
photoexcitation
(24, 101)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Hyperbolic plasmon-phonon
polaritons (graphene/hBN)
Fig. 1E 90 to 110 (type I),
170 to 200 (type II)
630 to 750 37 15 Gating,* doping,*
photoexcitation
(18, 60)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Hyperbolic plasmon-phonon
polaritons [Bi
2
Se
3
, (Bi,Sb)
2
Te
3
]
N/A 8 to 20 Gating,* doping,*
photoexcitation
(93)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Exciton polariton (WSe
2
) Fig. 1F 1400 to 1600 >300 3 5 Crystal thickness,* DE (45)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Hyperbolic phonon polariton
(h-BN)
Fig. 1G 90 to 110 (type I),
170 to 200 (type II)
200 to 1000 50 200 Crystal thickness,* DE* (17, 82, 102–104)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Topological SPP: Bi
2
Se
3
,
(BiSb)
2
Te
3
N/A <200 7 to 5000 900 3 Gating, doping* (105, 106)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Cooper-pair and Josephson
plasmon polaritons (cuprate
high-T
c
superconductors)
N/A <40 9000 to 13,000 3 4 Doping, crystal thickness,* DE* (89, 90, 107, 108)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Anisotropic SPP
(black phosphorus)
N/A <60 (k || G-X),
<40 (k || G-Y)
(109)
............ ................ ................ .................. ................ ................ ................ ............... ................ ................... ............... ................ ................ ................ ............... ................ ................... ............... ................ ................ ............
Fig. 1. Polaritons in vdW materials visualized through near-field nanoimaging. (A) Dirac pl asmons in graphene (20, 21). [Repr oduced from (20)] (B)Edge
plasmons at the boundary of a graphene nanor esonator (64, 65). [Repr oduced, with permission, from (65)] (C) One-dimensional plasmons in a carbon nanotube
(30). [Reproduced, with permission, from (30)] (D) Superlattice plasmons in graphene–h-BN moiré superlattices (24). [Reprod uced from (24)] (E)Hybrid
plasmon-phonon polaritons in graphene on h-BN (60). [Reproduced from (60)] (F) Ex citon po laritons (45)inWSe
2
. [Repr oduced, with permission, from (45)]
(G) Hyperbolic phonon polaritons in a h-B N slab (102), propagating as guided waves (schematic line).
RESEARCH | REVIEW
symmetric branch corresponds to the thin-
film plasmons (Eq. 1 and Fig. 3B). For k >> 1/d,
both of these branches are localized to the
film surfaces and are nearly degenerate. A set
of guided waves above the bulk plasmon fre-
quency w
p
may also exist between the light lines
of vac uu m and the material (tilted dashed lines
in Fig. 3B) for films with high-frequency permit-
tivity larger than unity.
Phonon and exciton polaritons
in dielectrics
A typical bulk dielectric exhibits three modes:
two transverse opt ical branches of phonon
po laritons (PhPs) w
TO
, generated by hybridiza-
tion of a photon (dashed line in Fig. 3C) and a
TO phonon; and one longitudinal phonon w
LO
,
analogous to the bulk plasmon in a metal. In a
semi-infinite dielectric, a surface phonon polar-
iton (SPhP) emerges inside the bulk stop band
w
TO
< w < w
LO
. In thin slabs (Fig. 3D), the PhP
branches spl it into guided modes while the SPhP
generates symmetric and antisymmetric modes
similar to SPPs in metal films. The mode struc-
ture of exciton polaritons in semiconductors is
similar to that of phonon polaritons in dielec-
trics, except that the role of w
TO
is played by
the exciton energy and the dispersion at large
momenta is quadratic: w(k)=w
TO
+(ħk
2
/2m
ex
),
where m
ex
is the exciton mass. The w
LO
-w
TO
gap
in excitonic systems is ref erred to as the Rabi
splitting.
Hyperbolic media and waveguide modes
Hyperbolic materials exhibit permittivities of
opposite signs along different directions. In
particular, type II hyperbolic materials possess
positive e
||
and negative e
⊥
.Inanisotropicpolar
dielectrics, this regime may be realized within
stop bands. Hyperbolicity leads to birefringence,
with the dispersion relatio ns of the ordinary and
extraordinary rays given by w
2
/c
2
= ðk
2
jj
þ k
2
⊥
Þ=e
⊥
and w
2
/c
2
= ðk
2
jj
=e
⊥
Þþðk
2
⊥
=e
jj
Þ, respectively. The
extraordinary rays have peculiar isofrequency
open surfaces, shaped as single-sheet hyper-
boloids (13–15). When projected on the w-k
⊥
plane, the hyperboloids fill a continuous region
(orange fringes in Fig. 3E). The transverse k
∥
and axial k
⊥
momenta of these extraordinary
polaritons can be very large, being limited only
by the atomic structure of the mater ial.
Polaritons can only propagate at angles q or
p – q with respect to the optical axis satisfying
the relation tan q =|e
⊥
/e
∥
|
1/2
.Thisimpliesthat
the polariton can be focused into nar row bea ms
that do not spread laterally as they propagate
through the material (see below). A thin slab
with surfaces normal to the optical axis sup-
ports weakly confined surface modes that evolve
into the principal branch of the guided waves
as k increases (Fig. 3F). This is accompanied by
numerous higher-order branches (Fig. 3F) that
arise from splitting of the extraordinary ray con-
tinuum in Fig. 3E. The group velocity can be neg-
a tive in a hyperbolic material, as demonstrated
by direct imaging (16, 17). Hyperbolic electro-
dynamics is ubiquitous in vdW materials and
originates not only in the phonon modes (Fig.
3, E and F) but also in a highly anisotropic
electronic response.
Plas mon-phonon polaritons are more com-
plex modes involving the hybridizat ion of the
corresponding elemental excitations in heter o-
structures. For example, in graphene supported
by hexagonal boron nitride (h-BN) (18, 60)
(Fig. 3E), the hyperbolic guided waves (13–15)
appear in the two bands marked type I and typ e
II. Outside these bands, one finds SPP and SP
branches similar to those in Fig. 3B. The slope
of the SP dispersion—the group velocity n
g
—is
much smaller than c: The light cone is nearly
vertical (and hence invisible) in Fig. 3G. Addi-
tionally, n
g
nearly everywhere exceeds the Fermi
velocity and the plasmon does not overlap the
electron-hole pair continuum (green region in
Fig. 3G), so Landau damping is preve nted.
New physics revealed
by polaritonic observations
Polaritons in vdW materials provide unique op-
portunities for exploring electronic phen omena
and lattice dynamics. In particular, polar itonic
images grant us access into regions of the dis-
persion relatio ns o f vari ous exc itation s beyond
what is attainable with conventional optics.
Interactions and many-body effects
The decay rate and wavelength of plasmonic and
polaritonic waves (Fig. 1) are determined by the
complex optical conductivity s (k, w)oftheme-
dium that supports these waves. It is thus possi-
ble to rec o ns t r u c t s(k, w) from polaritonic images,
which contain information on both electronic
and lattice dynamics (10, 19–21). Specifically, the
periodicity of plasmonic waves in graphene (Fig.
1A) is determined by the imaginary part of the
conductivity, whereas the rate at which these
waves decay into the interior of the samples is
governed by Re s/Im s. The plasmon propaga-
tion length ~(Im s/Re s)l
p
has been shown to
reach ~1 mm (i.e., tens of plasmon wavelengths) in
SCIENCE sciencemag.org 14 OCTO BER 20 16 • VOL 35 4 ISSUE 6309 aag1992-3
A
B
C
D
E
F
Fig. 2. Launching and visualizing polaritons. (A to F) Excitation and prob-
ing of vdW polaritons (blue arr ows) can be achiev ed using (A) periodic struc-
tures (110–112), (B) nonlinear wave mixing (95), (C) antenna-like nanotips
(20 , 21), (D) electron beams (113), (E) quantum dots and localiz ed emitters
(114, 115), and (F) elect r on tunneling (116). Polaritons produced b y processes
shown in (A) to (C) maintain phase coherence with respect to the external
illumination, in contr ast to mechanisms shown in (D) to (F), which are in-
elastic. A variant of (A) has been proposed that relies on surface acoustic wave
modulation (117, 118). Sample edges (64, 65) also provide additional momentum
to mediate light-polariton coupling. Localiz ed polaritons confined to nanoislands
can be re sonantly ex cited by incident light. Radiative outcoupling of polaritons
can be visualized by rever sing the arrows in (A) to (C).
RESEARCH | REVIEW