Accessing phonon polaritons in hyperbolic crystals by ARPES
Andrea Tomadin,
1
Alessandro Principi,
2
Justin C.W. Song,
3
Leonid S. Levitov,
4,
∗
and Marco Polini
1, 5,
†
1
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
2
Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
3
Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
4
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
5
Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
Recently studied hyperbolic materials host unique phonon-polariton (PP) modes. The ultra-
short wavelengths of these modes, which can be much smaller than those of conventional exciton-
polaritons, are of high interest for extreme sub-diffraction nanophotonics schemes. Polar hyperbolic
materials such as hexagonal boron nitride can be used to realize strong long-range coupling between
PP modes and extraneous charge degrees of freedom. The latter, in turn, can be used to control
and probe PP modes. Of special interest is coupling between PP modes and plasmons in an adja-
cent graphene sheet, which opens the door to accessing PP modes by angle-resolved photoemission
spectroscopy (ARPES). A rich structure in the graphene ARPES spectrum due to PP modes is
predicted, providing a new probe of PP modes and their coupling to graphene plasmons.
Introduction.—
The intrinsic hyperbolic character [1] of
hexagonal boron nitride (hBN) grants a unique platform
for realizing deep-subwavelength nanophotonic schemes.
Key to these developments are phonon-polariton (PP)
modes that exist within
reststrahlen
frequency bands [2,
3], characterized by wavelengths that can be as small as
1-100 nm. Highly directional, these modes exhibit deep
sub-diffraction confinement of light with wavelengths far
shorter than those of exciton-polaritons in semiconduc-
tor microcavities [4]. PPs have been shown to propagate
with low losses [2, 3] besting artificial metallic-resonator
metamaterial schemes, and opening the door to hyper-
lensing [5, 6].
Key to harnessing PP modes is gaining access to their
response over a wide wavenumber and energy band-
widths. However, to date PPs have only been studied
within a small frequency range limited by laser choice
(e.g.
167 meV
.
~
ω
.
198 meV via scattering-
type near-field optical spectroscopy technique [2]), or
at specific wavelengths fixed by the sample geometry
via Fourier transform infrared spectroscopy of nanofabri-
cated nanopillars [3]. New approaches allowing to resolve
the PP modes at shorter wavelengths and over a broad
range of energies are therefore highly desirable.
Here we describe an angle-resolved photoemission
spectroscopy (ARPES) [9] scheme to achieve broadband
energy-resolved access to ultra-short wavelength PPs in
hBN. At first glance, ARPES access to PPs in a wide-
bandgap insulator (hBN) where no free carriers are avail-
able may seem counterintuitive. However, the key to
our protocol lies in coupling PPs to charge degrees of
freedom in a conductor (e.g. graphene) placed nearby
the hyperbolic crystal of interest (hBN), prepared in a
slab geometry. Strong coupling [10–15] between hBN
Fabry-P ́erot PP modes and the collective charge oscilla-
tions (i.e. Dirac plasmons [16]) in a doped graphene sheet
placed over a hBN slab gives rise to new channels for
−
0
.
4
−
0
.
2
0
.
0
0
.
2
0
.
4
k
[nm
−
1
]
−
1
.
0
−
0
.
8
−
0
.
6
−
0
.
4
−
0
.
2
0
.
0
̄
hω
[eV]
(1)
(2)
0
.
0
0
.
5
1
.
0
1
.
5
2
.
0
FIG. 1. (Color online) Signatures of PP modes in the quasi-
particle spectral function
A
(
k
,ω
) of a doped graphene sheet
placed over a hBN slab, obtained from Eqs. (3) and (4). Note
the black linearly-dispersing quasiparticle bands, which dis-
play a clear Dirac crossing labeled by (1), and the broad spec-
tral feature labeled by (2) due to the emission of the plasmon-
phonon polariton mode with highest energy in Fig. 2(a). The
Fermi energy is positioned at
ω
= 0. Emission of polariton
modes [see Fig. 2(a)] by the holes created by photo-excited
electrons gives rise to four dispersive satellite bands running
parallel to the main quasiparticle bands (marked by red ar-
rows). The feature (2) is mainly plasmonic,whereas the satel-
lite bands, crossing at
k
= 0 between features (1) and (2), are
entirely due to Fabry-P ́erot hBN phonon-polariton modes.
Parameters used: Fermi energy
ε
F
= 400 meV, hBN slab
thickness
d
= 60 nm,
a
= 1 (vacuum),
b
= 3
.
9 (SiO
2
). The
colorbar refers to the values of
~
A
(
k
,ω
) in eV.
arXiv:1504.05345v1 [cond-mat.mes-hall] 21 Apr 2015
2
quasiparticle decay yielding a rich structure of dispersive
satellite features—marked by red arrows in Fig. 1—in the
graphene ARPES spectrum
A
(
k
,ω
). Since hBN Fabry-
P ́erot PPs are controlled by slab thickness, the composite
G/hBN structure features a novel ARPES spectrum with
features that are highly tunable by thickness.
The greatest practical advantage of this approach is
that ARPES achieves extreme resolution over a wide
range of wave vectors
k
(from the corners
K,K
′
of the
graphene Brillouin zone to the Fermi wave number
k
F
in graphene) and energies
~
ω
, with all energies below
the Fermi energy being probed simultaneously. This
gives an additonal benefit, besides tunability, in that
the entire range of frequencies and wavenumbers can
be covered within a single experiment. It is remarkable
that a one-atom-thick conducting material like graphene,
once placed over an insulating hyperbolic crystal, enables
ARPES studies of PP modes over the full range of wave
vectors and energies of interest.
From a more fundamental perspective, ARPES will
also be an ideal tool to investigate whether effective
electron-electron interactions mediated by the exchange
of PPs are capable of driving electronic systems towards
correlated states. Finally, looking at our results from the
point of view of graphene optoelectronics, one can en-
vision situations in which the tunable coupling between
graphene quasiparticles and the complex excitations of
its supporting substrate can be used to achieve control
over the
spectral
properties of graphene carriers, includ-
ing their decay rates, renormalized velocities, etc. This
degree of tunability may have important implications on
the performance of graphene-based photodetectors [17].
Phonon and plasmon-phonon polaritons.—
We consider
a vertical heterostructure—see inset in Fig. 2(b)—
composed of a graphene sheet located at
z
= 0 and placed
over a homogeneous anisotropic insulator of thickness
d
with dielectric tensor
ˆ
= diag(
x
,
y
,
z
). Homogeneous
and isotropic insulators with dielectric constants
a
and
b
fill the two half-spaces
z >
0 and
z <
−
d
, respec-
tively. The Fourier transform
V
q
,ω
of the Coulomb inter-
action potential, as dressed by the presence of a
uniaxial
(
y
=
x
) dielectric, is given by
V
q
,ω
=
φ
q
√
x
z
+
b
tanh(
qd
√
x
/
z
)
√
x
z
+ (
x
z
+
b
a
) tanh(
qd
√
x
/
z
)
/
(2 ̄
)
,
(1)
where
v
q
= 2
πe
2
/
(
q
̄
) with ̄
= (
a
+
b
)
/
2 is the ordi-
nary 2D Coulomb interaction potential. A more general
equation, which is also valid in the case
y
6
=
x
, can be
found in Sect. I of Ref. 18.
In the case of hBN, the components
x
and
z
of
the dielectric tensor have an important dependence on
frequency
ω
in the mid infrared [19].
The simplest
parametrization formulas for
x,z
=
x,z
(
ω
) are reported
in Sect. I of Ref. 18 and have been used for the numer-
ical calculations. More realistic parametrizations can be
found in the Supplementary Information of Ref. 13.
Standing PP modes [2] correspond to poles of the
dressed interaction
V
q
,ω
inside
the reststrahlen bands.
These can be found by looking at the zeroes of the denom-
inator in Eq. (1),
√
|
x
(
ω
)
z
(
ω
)
|
+ (2 ̄
)
−
1
[
x
(
ω
)
z
(
ω
) +
b
a
] tan[
qd
√
|
x
(
ω
)
/
z
(
ω
)
|
] = 0. Illustrative numerical
results for
d
= 10 nm and
d
= 60 nm are reported in
Fig. 1 of Ref. 18. Analytical expressions, which are valid
for
qd
1 and
qd
1, are available [18] in the case
in which phonon losses in hBN are neglected. For suffi-
ciently thick hBN slabs, there can be modes with group
velocity equal to the graphene Fermi velocity
v
F
.
Standing PP modes in a hBN slab couple to Dirac plas-
mons in a nearby graphene sheet. Such coupling is cap-
tured by the random phase approximation (RPA) [20].
In the RPA, one introduces the dynamically screened in-
teraction
W
q
,ω
=
V
q
,ω
ε
(
q
,ω
)
≡
V
q
,ω
1
−
V
q
,ω
χ
0
(
q,ω
)
.
(2)
Here
ε
(
q
,ω
) is the RPA dielectric function and
χ
0
(
q,ω
)
is the density-density response function of a 2D massless
Dirac fermion fluid [21]. While the poles of
V
q
,ω
physi-
cally yield slab PP modes, new poles of
W
q
,ω
emerge from
electron-phonon interactions. These are weakly-damped
solutions
ω
= Ω
q
−
i
0
+
of the equation
ε
(
q
,ω
) = 0.
We have solved this equation numerically and illustra-
tive results for
ε
F
= 400 meV and
d
= 60 nm are shown
in Fig. 2(a). (Results for different values of
ε
F
and
d
can be found in Sect. III of Ref. 18.) Solid lines repre-
sent plasmon-phonon polaritons that emerge from the hy-
bridization between the Dirac plasmon [16] in graphene
(dashed line) and standing PP waves in the hBN slab.
The solid red lines denote three polariton branches with
a strong degree of plasmon-phonon hybridization. On
the contrary, black solid lines denote practically unhy-
bridized slab PP modes. We clearly see that there are
several plasmon-phonon polariton modes (green circles)
with group velocity equal to
v
F
. These modes couple
strongly to quasiparticles in graphene, as we now pro-
ceed to demonstrate.
Quasiparticle decay rates.—
An excited quasiparticle with
momentum
k
and energy
~
ω
, created in graphene in
an ARPES experiment [22–26], can decay by scattering
against the excitations of the Fermi sea, i.e. electron-hole
pairs and collective modes. The decay rate
~
/τ
λ
(
k
,ω
) for
these processes can be calculated [20] from the imaginary
part of the retarded quasiparticle self-energy Σ
λ
(
k
,ω
),
i.e.
~
/τ
λ
(
k
,ω
) =
−
2Im [Σ
λ
(
k
,ω
)]. In the RPA and at
zero temperature we have [27, 28]
Im [Σ
λ
(
k
,ω
)] =
∑
λ
′
∫
d
2
q
(2
π
)
2
Im
[
W
q
,ω
−
ξ
λ
′
,
k
+
q
]
F
λλ
′
×
[Θ(
~
ω
−
ξ
λ
′
,
k
+
q
)
−
Θ(
−
ξ
λ
′
,
k
+
q
)]
.
(3)
Here
F
λλ
′
≡
[1 +
λλ
′
cos (
θ
k
,
k
+
q
)]
/
2 is the chirality fac-
tor [27, 28],
ξ
λ,
k
=
λ
~
v
F
k
−
ε
F
is the Dirac band energy
3
measured from the Fermi energy
ε
F
(
λ,λ
′
=
±
1), and
Θ(
x
) is the usual Heaviside step function. The quantity
~
ω
is also measured from the Fermi energy and, finally,
θ
k
,
k
+
q
is the angle between
k
and
k
+
q
. Eq. (3) reduces
to the standard Fermi golden rule when only terms of
O
(
V
2
q
,ω
) are retained. Physically, it describes the decay
rate of a process in which an initial state with momen-
tum
k
and energy
~
ω
(measured from
ε
F
) decays into
a final state with momentum
k
+
q
and energy
ξ
λ
′
,
k
+
q
(measured from
ε
F
). For
ω <
0, the self-energy expresses
the decay of
holes
created inside the Fermi sea, which
scatter to a final state, by exciting the Fermi sea. Fermi
statistics requires the final state to be
occupied
so both
band indices
λ
′
=
±
1 are allowed in the case
ε
F
>
0 that
we consider here. Since ARPES measures the properties
of holes produced in the Fermi sea by photo-ejection,
only
ω <
0 is relevant for this experimental probe in an
n
-doped graphene sheet.
It is convenient to discuss the main physical features
of Im [Σ
λ
(
k
,ω
)] for an initial hole state with momen-
tum
k
=
0
. In this case, the 2D integral in Eq. (3)
reduces to a simple 1D quadrature. The initial hole
energy is
E
i
=
~
ω
+
ε
F
.
The final hole energy is
E
f
=
ξ
λ
′
,
q
+
ε
F
=
λ
′
~
v
F
q
. When the difference ∆
λ
′
,q
≡
E
f
−
E
i
is equal to the real part of the mode energy
~
Ω
q
, the initial hole, which has been left behind after
the photo-ejection of an electron, can decay by emit-
ting a plasmon-phonon polariton. Since
~
Ω
q
>
~
v
F
q
,
but ∆
λ
′
,q
≤
~
v
F
q
for intraband transitions, an initial
hole state with
E
i
<
0 (i.e. initial hole state in valence
band) can decay only into a final hole state with
E
f
>
0
(i.e. final hole state in conduction band). In particular,
when
d
Ω
q
/dq
=
~
−
1
d
∆
λ
′
,q
/dq
=
λ
′
v
F
, such decay pro-
cess is
resonant
. When these conditions are met, the
inter-band contribution to Im [Σ
λ
(
0
,ω
)] peaks at a char-
acteristic value of
ω
and the Kramers-Kronig transform
Re[Σ
λ
(
0
,ω
)] changes sign rapidly around that frequency.
Within RPA, a satellite quasiparticle emerges [29], which
is composed by a hole that moves with the same speed
of a plasmon-phonon polariton. This is a solution of the
Dyson equation, distinct from the ordinary quasiparticle
solution that becomes the Landau pole of the one-body
Green’s function as
k
→
k
F
and
ω
→
0.
The quantity Im [Σ
λ
(
0
,ω
)], calculated from Eq. (3), is
plotted as a function of
ω
in Fig. 2(b), for
ε
F
= 400 meV
and
d
= 60 nm. (The dependence of the decay rate
on
ε
F
and
d
is discussed in Sect. III of Ref. 18.) We
clearly see several peaks in Im [Σ
λ
(
0
,ω
)] for
~
ω <
−
ε
F
(
E
i
<
0), which occur at values of
~
ω
that are in a
one-to-one correspondence with the “resonant” plasmon-
phonon polaritons, i.e. polaritons with group velocity
equal to
v
F
, shown in Fig. 2(a). Indeed, as stated above,
peaks in Im [Σ
λ
(
0
,ω
)] are expected at values of
~
ω
—
marked by green vertical lines in Fig. 2(b)—given by
~
ω
=
~
v
F
q
?
−
ε
F
−
~
Ω
q
?
, where
q
?
is the wave num-
ber at which the resonance condition
d
Ω
q
/dq
=
v
F
is
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
q
[nm
−
1
]
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
̄
hω
[eV]
(a)
−
2
.
0
−
1
.
5
−
1
.
0
−
0
.
5
0
.
0
̄
hω/ε
F
0
.
01
0
.
1
1
10
−=
m
[Σ
λ
(
0
,ω
)]
/ε
F
(b)
−
2
.
0
−
1
.
5
−
1
.
0
−
0
.
5
0
.
0
̄
hω/ε
F
0
.
00
0
.
05
0
.
10
0
.
15
0
.
20
−=
m
[Σ
λ
(
0
,ω
)]
/ε
F
(c)
FIG. 2.
(Color online) Panel (a) Dispersion relation Ω
q
of
hybrid plasmon-phonon polaritons (solid lines) with param-
eters as in Fig. 1. The dashed line represents the dispersion
relation of a Dirac plasmon [16] in graphene, in the absence
of hBN phonons. Horizontal cyan areas denote the hBN rest-
strahlen bands. The grey-shaded area represents the intra-
band particle-hole continuum in graphene. Green filled cir-
cles represent the points where the plasmon-phonon polariton
group velocity equals the graphene Fermi velocity
v
F
. Panel
(b) The quantity
−
Im[Σ
λ
(
k
,ω
)] (in units of
ε
F
and evaluated
at
k
=
0
) is shown as a function of the rescaled frequency
~
ω/ε
F
. Green vertical lines denote the values of
~
ω/ε
F
at
which a plasmon-phonon polariton peak is expected. The ver-
tical axis is in logarithmic scale. The inset shows a side view of
the vertical heterostructure analyzed in this work. Panel (c)
Same as in panel (b) but in the absence of dynamical screening
due to electron-electron interactions in graphene: these nu-
merical results have been obtained by replacing
W
q
,ω
→
V
q
,ω
in Eq. (3). A polaron peak is clearly visible.
satisfied. For example, the resonant mode at highest en-
ergy in Fig. 2(a), which occurs at
q
?
≈
0
.
26 nm
−
1
and
energy
~
Ω
q
?
≈
0
.
36 eV, yields a peak in Im [Σ
λ
(
0
,ω
)] at
~
ω/ε
F
≈−
1
.
5, see Fig. 2(b).
Comparing Fig. 2(b) with Fig. 2(c), we clearly see the
4
−
800
−
700
−
600
−
500
−
400
̄
hω
[meV]
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
̄
h
A
(
0
,ω
) [eV]
d
= 10 nm
(a)
−
575
−
570
−
565
−
560
−
555
̄
hω
[meV]
30
40
50
60
70
80
90
100
̄
h
A
(
0
,ω
) [meV]
d
(b)
−
480
−
478
−
476
−
474
−
472
−
470
̄
hω
[meV]
0
.
0
0
.
5
1
.
0
1
.
5
2
.
0
̄
h
A
(
0
,ω
)) [eV]
d
(c)
FIG. 3. (Color online) Panel (a) The quasiparticle spectral
function
A
(
k
,ω
) evaluated at
k
=
0
(
|
k
|
= 10
−
3
k
F
has been
used in the numerical calculations) is plotted as a function of
~
ω
. This plot refers to
d
= 10 nm. The other parameters are
as in Fig. 1. Panels (b) and (c) Dependence on the hBN slab
thickness
d
of the spectral function features highlighted by
vertical orange-shaded regions in panel (a). Different curves
correspond to values of
d
on a uniform mesh from
d
= 10 nm
to
d
= 60 nm. Arrows indicate how spectral features evolve
by increasing
d
.
role of dynamical screening due to electron-electron inter-
actions in graphene. For
ε
(
k
,ω
) = 1, the off-shell decay
rate Im [Σ
λ
(
0
,ω
)] shows only a polaron peak, due to the
emission of a Fabry-Per ́ot PP mode with group velocity
equal to
v
F
, see Fig. 1 in Ref. 18.
At
k
6
=
0
, the conduction and valence band
Im[Σ
λ
(
k
,ω
)] plasmon-phonon polariton peaks broaden
and separate, because of [27] the dependence on scatter-
ing angle of
ξ
λ
′
,
k
+
q
and the chirality factor
F
λλ
′
, which
emphasizes
k
and
q
in nearly parallel directions for con-
duction band states and
k
and
q
in nearly opposite direc-
tions for valence band states. As a result, the conduction
band plasmon-phonon polariton peak moves up in energy
while the valence band peak moves down.
Quasiparticle spectral function.—
An ARPES exper-
iment [9] probes the quasiparticle spectral function
A
(
k
,ω
) =
−
π
−
1
∑
λ
Im[
G
λ
(
k
,ω
)] =
∑
λ
=
±
1
A
λ
(
k
,ω
)
of the occupied states below the Fermi energy. Here
G
λ
(
k
,ω
) is the one-body Green’s function in the band
representation and
A
λ
=
−
1
π
ImΣ
λ
(
ω
−
ξ
λ,
k
/
~
−
ReΣ
λ
/
~
)
2
+ (ImΣ
λ
/
~
)
2
.
(4)
In writing Eq. (4) we have dropped explicit reference
to the
k
,ω
variables.
The real part Re[Σ
λ
(
k
,ω
)] of
the quasiparticle self-energy can be calculated, at least
in principle, from the Kramers-Kronig transform of
Im[Σ
λ
(
k
,ω
)]. A more convenient way to handle the
numerical evaluation of Re[Σ
λ
(
k
,ω
)] is to employ the
Quinn-Ferrell line-residue decomposition [30].
Our main results for the quasiparticle spectral func-
tion
A
(
k
,ω
) of a doped graphene sheet placed on a hBN
slab are summarized in Fig. 1 and Fig. 3. We clearly
see that the presence of the hBN substrate is respon-
sible for the appearance of a family of sharp dispersive
satellite features associated with the presence of PPs and
plasmon-phonon polaritons. This is particularly clear in
the one-dimensional cut at
k
=
0
of
A
(
k
,ω
) displayed
in Fig. 3(a) for
d
= 10 nm. All the sharp structures
between the ordinary quasiparticle peak slightly below
~
ω
=
−
0
.
4 eV and the peak at
~
ω
≈ −
0
.
7 eV, which
is mostly plasmonic in nature, are sensitive to the de-
tailed distribution and dispersion of Fabry-Per ́ot PP in
the hBN slab, and therefore to the slab thickness
d
. This
is clearly shown in Fig. 3(b) and (c), where we see shifts
of these peaks of several meV, when
d
is changed from
d
= 10 nm to
d
= 60 nm, while keeping
ε
F
constant.
In summary, we have studied the coupling between
standing phonon-polariton modes in a hyperbolic crys-
tal slab and the plasmons of the two-dimensional mass-
less Dirac fermion liquid in a nearby graphene sheet.
We have shown that this coupling yields a complex
spectrum of (plasmon-phonon) polaritons, see Fig. 2(a).
Plasmon-phonon polaritons with group velocity equal
to the graphene Fermi velocity couple strongly with
graphene quasiparticles, enabling ARPES access to PP
modes in hyperbolic crystal slabs, as shown in Figs. 1
and 3. Recent progress [31] in the chemical vapor de-
position growth of large-area graphene/hBN stacks on
Cu(111) in ultrahigh vacuum and the ARPES character-
ization of the resulting samples makes us very confident
on the observability of our predictions. Our findings sug-
gest that appropriate coupling of graphene to substrates
which allow strong plasmon-phonon hybridization could
open the route to the manipulation of carriers’ spectral
properties, paving the way for novel device functionali-
ties.
Acknowledgements.—
We gratefully acknowledge F.H.L.
Koppens for useful discussions. This work was supported
5
by the EC under the Graphene Flagship program (con-
tract no. CNECT-ICT-604391) (A.T. and M.P.), MIUR
(A.T. and M.P.) through the programs “FIRB - Futuro
in Ricerca 2010” - Project “PLASMOGRAPH” (Grant
No. RBFR10M5BT) and “Progetti Premiali 2012” -
Project “ABNANOTECH”, the U.S. Department of En-
ergy under grant DE-FG02-05ER46203 (A.P.), and a Re-
search Board Grant at the University of Missouri (A.P.).
Work at MIT was supported as part of the Center for
Excitonics, an Energy Frontier Research Center funded
by the U.S. Department of Energy, Office of Science, Ba-
sic Energy Sciences under Award No. desc0001088. This
work was also supported, in part, by the U.S. Army Re-
search Laboratory and the U.S. Army Research Office
through the Institute for Soldier Nanotechnologies, un-
der contract number W911NF-13-D-0001. Free software
(www.gnu.org, www.python.org) was used.
∗
levitov@mit.edu
†
marco.polini@icloud.com
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