Nonlinear Interactions between Free Electrons
and Nanographenes
Joel D. Cox
†,‡
and F. Javier García de Abajo
∗,¶,§
†Center for Nano Optics, University of Southern Denmark, Campusvej 55, DK-5230
Odense M, Denmark
‡Danish Institute for Advanced Study, University of Southern Denmark, Campusvej 55,
DK-5230 Odense M, Denmark
¶ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels (Barcelona), Spain
§ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23,
08010 Barcelona, Spain
E-mail: javier.garciadeabajo@nanophotonics.es
Abstract
Free electrons act as a source of highly confined, spectrally broad op-
tical fields that are widely used to map photonic modes with nanome-
ter/millielectronvolt space/energy resolution through currently available
electron energy-loss and cathodoluminescence spectroscopies. These tech-
niques are understood as probes of the linear optical response, while non-
linear dynamics has escaped observation with similar degree of spatial de-
tail, despite the strong enhancement of the electron evanescent field with
decreasing electron energy. Here, we show that the field accompanying
low-energy electrons can trigger anharmonic response in strongly nonlinear
1
materials. Specifically, through realistic quantum-mechanical simulations,
we find that the interaction between . 100 eV electrons and plasmons in
graphene nanostructures gives rise to substantial optical nonlinearities that
are discernable as saturation and spectral shifts in the plasmonic features
revealed in the cathodoluminescence emission and electron energy-loss spec-
tra. Our results support the use of low-energy electron-beam spectroscopies
for the exploration of nonlinear optical processes in nanostructures.
Keywords: nonlinear optics, electron beams, graphene plasmons, electron microscopy,
nanographenes, cathodoluminescence
Optical spectroscopies rely on the response of the sample to the electromagnetic field of
an external light source. When combined with far-field microscopy, the spatial resolution
of these techniques is limited by diffraction to roughly half the light wavelength,
1
while the
use of tips in scanning near-field optical setups permits imaging features with sizes down to
tens of nanometers.
2
A substantial gain in spatial resolution can be achieved if the exter-
nal field is supplied by electron beams (e-beams), which can be currently focused down to
sub-Ångstrom spots, where spectral analysis of the energy losses experienced by the elec-
trons
3–5
or the cathodoluminescence (CL) light emission resulting from their interaction with
the sample
6
allow us to identify optical excitations with millielectronvolt energy resolution.
In an intuitive picture, the passage of an electron produces a transient evanescent electric
field that can be regarded as a broad optical pulse capable of exciting the sample. In CL
spectroscopy, an optical monochromator separates different frequency components of the
scattered far-field (i.e., different emitted photon energies) resulting from that interaction,
while in electron energy-loss spectroscopy (EELS) an electron analyzer is used to resolve the
excitation frequencies as spectral features associated with energy losses experienced by the
transmitted electrons. These techniques have been extensively applied to study plasmons
in nanostructures,
7–12
optical modes in photonic crystals,
13–15
and more recently, localized
phonon polaritons in the mid-infrared spectral range.
3–5
The interaction of e-beams with
2
engineered photonic structures has been also explored as a mechanism for integrated light
sources.
16,17
Adding to the unprecedented combination of space and energy resolution en-
abled by electron beams, ultrafast temporal resolution has been achieved through the use
of ultrashort electron pulses emitted from a cathode under femtosecond laser pulse irradia-
tion.
18–20
Because the excitation yield of optical modes by e-beams is generally found to be small
(e.g., < 10
−4
per electron at typical beam energies ∼ 100 keV), EELS and CL are commonly
regarded as probes of the linear optical response. Nevertheless, the amplitude of the evanes-
cent field provided by a moving electron at a specific frequency (i.e., the spectral component
of that field in the frequency range of the sampled mode) scales as ∼ 1/v at low velocity
v c,
6
thus resulting in a ∼ 1/v
2
dependence of the excitation yield. In fact, for < 100 eV
electrons and strong plasmonic optical modes, the interaction has been theoretically shown
to reach unity order.
21
We thus expect that low-energy e-beams can trigger an anharmonic
response in strongly nonlinear nanostructures such as graphene nanoislands, where a depar-
ture from the linear regime should be already observable at the level of a single plasmon
excitation.
22
Here, we demonstrate through realistic quantum-mechanical simulations that low-energy
e-beams can indeed trigger nonlinear optical response in graphene nanoislands, giving rise
to saturation and sizeable frequency shifts of the peaks in CL emission and energy loss
spectra associated with the excitation of plasmons in the sample. We base these results on
time-domain simulations of the electron-graphene interaction, with the latter described in a
self-consistent field approximation that incorporates nonlinear processes at all orders, and
the electron introduced as the external field produced by a moving point charge. Nonlinear
effects are revealed by studying the dependence of CL and energy loss spectra on the charge of
the probe q, which shows a clear departure from the linear regime (i.e., quadratic dependence
on q) already for q = −e (one electron). In a practical scenario, we find that nonlinear effects
can be probed by studying the dependence of the emission or energy loss spectra on e-beam
3
energy and lateral position, where spectral shifts as large as the plasmon linewidth should be
feasible under optimal conditions. The present study supports the potential of low-energy
electrons to explore the nonlinear optical response of nanostructures with unprecedented
spatial resolution.
0.1 101
10
7
10
8
10
0
10
-1
10
-2
10
-3
10
-4
Distance, size (nm)
Equivalent fluence (J/m
2
)
Equivalent light field (V/m)
(a)
10
6
gold spheres
e-beam
electron
Coulomb
interaction
plasmon
CL light
emission
(b)
Figure 1: Nonlinear optics with electron beams. (a) A free electron passing near
a nanostructure can interact multiple times via its evanescent Coulomb field with one of
the sample plasmons, triggering nonlinear optical response that in turn leaves a signature
in the resulting cathodoluminescence (CL) light emission. (b) The strength of the field
produced by the electron depends on distance to its trajectory and has an effect on a sample
resonance of lifetime τ similar to a spectrally narrow light pulse with an equivalent fluence
F
eff
≈ e
2
c/(2πv
2
R
2
τ) (left scale) and electric field strength E = (2e/vR)/
√
τ∆ (right scale),
as shown in this plot for 25 eV electron energy (v ≈ c/100), ~τ
−1
= 10 meV (τ ≈ 66 fs), and
∆ = 100 fs light pulse duration. Estimates of the onsets of nonlinear response in self-standing
gold spheres and silicon-supported graphene disks are shown for comparison as a function of
their size (see main text).
Equivalent Optical Pulse Fluence of a Free Electron. Electrons moving in vacuum
produce an evanescent electromagnetic field that can interact with the optical modes of a
sample, giving rise to energy losses and emission of radiation
6
(Figure 1a). This is the basis of
the EELS and CL techniques, the spectra of which are conveniently studied by time-Fourier
transforming the electric field produced by the electron, E
ext
(r, t) =
R
(dω/2π)e
−iωt
E
ext
(r, ω).
For constant nonrelativistic velocity v c, the frequency-space electric field intensity reduces
to
6
|E
ext
(r, ω)|
2
= (2eω/v
2
)
2
K
2
m
(ωR/v) as a function of distance R normal to the trajectory,
where K
m
are modified Bessel functions and m = 0 (m = 1) must be selected for directions
parallel (perpendicular) to the velocity vector v. For low photon energies ~ω . 1 eV and
4
small distances (R ∼few nm), the perpendicular component becomes dominant and the
Bessel function in this expression can be approximated as K
1
(θ) ≈ 1/θ, even for electron
velocities as small as v ∼ c/100 (i.e., ∼ 25 eV electrons), leading to a frequency-independent
field amplitude 2e/(vR).
The transient evanescent field of the electron presents a fluence F = (c/2π)
R
dt|E
ext
(r, t)|
2
=
(c/4π
2
)
R
dω |E
ext
(r, ω)|
2
. In order to estimate the effect of this field on sample excitations,
we consider an optical mode of frequency ω
0
and lifetime τ , which allows us to define an
effective fluence F
eff
= (c/4π
2
)
R
dω |E
ext
(r, ω)|
2
/[1+4(ω −ω
0
)
2
τ
2
] obtained by weighting the
spectral field components with the corresponding Lorentzian line shape of the resonance. As
we argue above, E
ext
(r, ω) is nearly independent of ω for the low resonance energies and small
distances under consideration (ω
0
R v), thus allowing us to obtain F
eff
≈ e
2
c/(2πv
2
R
2
τ)
(Figure 1b, blue curve). Now, it is convenient to compare this estimate with a typical ul-
trashort Gaussian light pulse (2Re{E e
−iω
0
t−t
2
/∆
2
} field profile) of ∆ = 100 fs duration and
the same fluence cE
2
∆/8π = F
eff
; this leads to an equivalent light field peak amplitude
E ∼ 10
7
V/m at a distance R = 1 nm for 25 eV electrons acting on an excitation of width
~τ
−1
= 10 meV (Figure 1b, blue line and right vertical axis), which is sufficient to produce
substantial nonlinear effects in several materials, and in particular, gold and graphene, as
we discuss below.
Highly doped graphene inherits a strong nonlinear response from its conical electronic
band structure with constant Fermi velocity v
F
≈ c/300.
23–25
Additionally, this material ex-
hibits electrically tunable infrared plasmons
26–33
that have been argued to boost the nonlinear
response to unpredecedented levels.
34
Following a previously reported analytical description
of plasmon-mediated nonlinear response of graphene islands,
35
in excellent agreement with
quantum-mechanical calculations at the same level of theory used in the present work, we
can readily quantify the external optical field amplitude E that needs to be applied to a
nanographene structure in order to produce a nonlinear dipole strength similar to the linear
one; here, we present a succinct description of the procedure described in the Methods sec-
5
